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10.10: Chapter 8 Review Exercises


Chapter Review Exercises

Simplify Expressions with Roots

Exercise (PageIndex{1}) Simplify Expressions with Roots

In the following exercises, simplify.

    1. (sqrt{225})
    2. (-sqrt{16})
    1. (-sqrt{169})
    2. (sqrt{-8})
    1. (sqrt[3]{8})
    2. (sqrt[4]{81})
    3. (sqrt[5]{243})
    1. (sqrt[3]{-512})
    2. (sqrt[4]{-81})
    3. (sqrt[5]{-1})
Answer

1.

  1. (15)
  2. (-4)

3.

  1. (2)
  2. (3)
  3. (3)

Exercise (PageIndex{2}) Estimate and Approximate Roots

In the following exercises, estimate each root between two consecutive whole numbers.

    1. (sqrt{68})
    2. (sqrt[3]{84})
Answer

1.

  1. (8
  2. (4

Exercise (PageIndex{3}) Estimate and Approximate Roots

In the following exercises, approximate each root and round to two decimal places.

    1. (sqrt{37})
    2. (sqrt[3]{84})
    3. (sqrt[4]{125})
Answer

1. Solve for yourself

Exercise (PageIndex{4}) Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

    1. (sqrt[3]{a^{3}})
    2. (sqrt[7]{b^{7}})
    1. (sqrt{a^{14}})
    2. (sqrt{w^{24}})
    1. (sqrt[4]{m^{8}})
    2. (sqrt[5]{n^{20}})
    1. (sqrt{121 m^{20}})
    2. (-sqrt{64 a^{2}})
    1. (sqrt[3]{216 a^{6}})
    2. (sqrt[5]{32 b^{20}})
    1. (sqrt{144 x^{2} y^{2}})
    2. (sqrt{169 w^{8} y^{10}})
    3. (sqrt[3]{8 a^{51} b^{6}})
Answer

1.

  1. (a)
  2. (|b|)

3.

  1. (m^{2})
  2. (n^{4})

5.

  1. (6a^{2})
  2. (2b^{4})

Simplify Radical Expressions

Exercise (PageIndex{5}) Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

  1. (sqrt{125})
  2. (sqrt{675})
    1. (sqrt[3]{625})
    2. (sqrt[6]{128})
Answer

1. (5sqrt{5})

3.

  1. (5 sqrt[3]{5})
  2. (2 sqrt[6]{2})

Exercise (PageIndex{6}) Use the Product Property to Simplify Radical Expressions

In the following exercises, simplify using absolute value signs as needed.

    1. (sqrt{a^{23}})
    2. (sqrt[3]{b^{8}})
    3. (sqrt[8]{c^{13}})
    1. (sqrt{80 s^{15}})
    2. (sqrt[5]{96 a^{7}})
    3. (sqrt[6]{128 b^{7}})
    1. (sqrt{96 r^{3} s^{3}})
    2. (sqrt[3]{80 x^{7} y^{6}})
    3. (sqrt[4]{80 x^{8} y^{9}})
    1. (sqrt[5]{-32})
    2. (sqrt[8]{-1})
    1. (8+sqrt{96})
    2. (frac{2+sqrt{40}}{2})
Answer

2.

  1. (4left|s^{7} ight| sqrt{5 s})
  2. (2 a sqrt[5]{3 a^{2}})
  3. (2|b| sqrt[6]{2 b})

4.

  1. (-2)
  2. not real

Exercise (PageIndex{7}) Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

    1. (sqrt{frac{72}{98}})
    2. (sqrt[3]{frac{24}{81}})
    3. (sqrt[4]{frac{6}{96}})
    1. (sqrt{frac{y^{4}}{y^{8}}})
    2. (sqrt[5]{frac{u^{21}}{u^{11}}})
    3. (sqrt[6]{frac{v^{30}}{v^{12}}})
  1. (sqrt{frac{300 m^{5}}{64}})
    1. (sqrt{frac{28 p^{7}}{q^{2}}})
    2. (sqrt[3]{frac{81 s^{8}}{t^{3}}})
    3. (sqrt[4]{frac{64 p^{15}}{q^{12}}})
    1. (sqrt{frac{27 p^{2} q}{108 p^{4} q^{3}}})
    2. (sqrt[3]{frac{16 c^{5} d^{7}}{250 c^{2} d^{2}}})
    3. (sqrt[6]{frac{2 m^{9} n^{7}}{128 m^{3} n}})
    1. (frac{sqrt{80 q^{5}}}{sqrt{5 q}})
    2. (frac{sqrt[3]{-625}}{sqrt[3]{5}})
    3. (frac{sqrt[4]{80 m^{7}}}{sqrt[4]{5 m}})
Answer

1.

  1. (frac{6}{7})
  2. (frac{2}{3})
  3. (frac{1}{2})

3. (frac{10 m^{2} sqrt{3 m}}{8})

5.

  1. (frac{1}{2|p q|})
  2. (frac{2 c d sqrt[5]{2 d^{2}}}{5})
  3. (frac{|m n| sqrt[6]{2}}{2})

Simplify Rational Exponents

Exercise (PageIndex{8}) Simplify Expressions with (a^{frac{1}{n}})

In the following exercises, write as a radical expression.

    1. (r^{frac{1}{2}})
    2. (s^{frac{1}{3}})
    3. (t^{frac{1}{4}})
Answer

1.

  1. (sqrt{r})
  2. (sqrt[3]{s})
  3. (sqrt[4]{t})

Exercise (PageIndex{9}) Simplify Expressions with (a^{frac{1}{n}})

In the following exercises, write with a rational exponent.

    1. (sqrt{21p})
    2. (sqrt[4]{8q})
    3. (4sqrt[6]{36r})
Answer

1. Solve for yourself

Exercise (PageIndex{10}) Simplify Expressions with (a^{frac{1}{n}})

In the following exercises, simplify.

    1. (625^{frac{1}{4}})
    2. (243^{frac{1}{5}})
    3. (32^{frac{1}{5}})
    1. ((-1,000)^{frac{1}{3}})
    2. (-1,000^{frac{1}{3}})
    3. ((1,000)^{-frac{1}{3}})
    1. ((-32)^{frac{1}{5}})
    2. ((243)^{-frac{1}{5}})
    3. (-125^{frac{1}{3}})
Answer

1.

  1. (5)
  2. (3)
  3. (2)

3.

  1. (-2)
  2. (frac{1}{3})
  3. (-5)

Exercise (PageIndex{11}) Simplify Expressions with (a^{frac{m}{n}})

In the following exercises, write with a rational exponent.

    1. (sqrt[4]{r^{7}})
    2. ((sqrt[5]{2 p q})^{3})
    3. (sqrt[4]{left(frac{12 m}{7 n} ight)^{3}})
Answer

1. Solve for yourself

Exercise (PageIndex{12}) Simplify Expressions with (a^{frac{m}{n}})

In the following exercises, simplify.

    1. (25^{frac{3}{2}})
    2. (9^{-frac{3}{2}})
    3. ((-64)^{frac{2}{3}})
    1. (-64^{frac{3}{2}})
    2. (-64^{-frac{3}{2}})
    3. ((-64)^{frac{3}{2}})
Answer

1.

  1. (125)
  2. (frac{1}{27})
  3. (16)

Exercise (PageIndex{13}) Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

    1. (6^{frac{5}{2}} cdot 6^{frac{1}{2}})
    2. (left(b^{15} ight)^{frac{3}{5}})
    3. (frac{w^{frac{2}{7}}}{w^{frac{9}{7}}})
    1. (frac{a^{frac{3}{4}} cdot a^{-frac{1}{4}}}{a^{-frac{10}{4}}})
    2. (left(frac{27 b^{frac{2}{3}} c^{-frac{5}{2}}}{b^{-frac{7}{3}} c^{frac{1}{2}}} ight)^{frac{1}{3}})
Answer

1.

  1. (6^{3})
  2. (b^{9})
  3. (frac{1}{w})

Add, Subtract and Multiply Radical Expressions

Exercise (PageIndex{14}) add and Subtract Radical Expressions

In the following exercises, simplify.

    1. (7 sqrt{2}-3 sqrt{2})
    2. (7 sqrt[3]{p}+2 sqrt[3]{p})
    3. (5 sqrt[3]{x}-3 sqrt[3]{x})
    1. (sqrt{11 b}-5 sqrt{11 b}+3 sqrt{11 b})
    2. (8 sqrt[4]{11 c d}+5 sqrt[4]{11 c d}-9 sqrt[4]{11 c d})
    1. (sqrt{48}+sqrt{27})
    2. (sqrt[3]{54}+sqrt[3]{128})
    3. (6 sqrt[4]{5}-frac{3}{2} sqrt[4]{320})
    1. (sqrt{80 c^{7}}-sqrt{20 c^{7}})
    2. (2 sqrt[4]{162 r^{10}}+4 sqrt[4]{32 r^{10}})
  1. (3 sqrt{75 y^{2}}+8 y sqrt{48}-sqrt{300 y^{2}})
Answer

1.

  1. (4sqrt{2})
  2. (9sqrt[3]{p})
  3. (2sqrt[3]{x})

3.

  1. (7sqrt{3})
  2. (7sqrt[3]{2})
  3. (3sqrt[4]{5})

5. (37 y sqrt{3})

Exercise (PageIndex{15}) Multiply Radical Expressions

In the following exercises, simplify.

    1. ((5 sqrt{6})(-sqrt{12}))
    2. ((-2 sqrt[4]{18})(-sqrt[4]{9}))
    1. (left(3 sqrt{2 x^{3}} ight)left(7 sqrt{18 x^{2}} ight))
    2. (left(-6 sqrt[3]{20 a^{2}} ight)left(-2 sqrt[3]{16 a^{3}} ight))
Answer

2.

  1. (126 x^{2} sqrt{2})
  2. (48 a sqrt[3]{a^{2}})

Exercise (PageIndex{16}) Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

    1. (sqrt{11}(8+4 sqrt{11}))
    2. (sqrt[3]{3}(sqrt[3]{9}+sqrt[3]{18}))
    1. ((3-2 sqrt{7})(5-4 sqrt{7}))
    2. ((sqrt[3]{x}-5)(sqrt[3]{x}-3))
  1. ((2 sqrt{7}-5 sqrt{11})(4 sqrt{7}+9 sqrt{11}))
    1. ((4+sqrt{11})^{2})
    2. ((3-2 sqrt{5})^{2})
  2. ((7+sqrt{10})(7-sqrt{10}))
  3. ((sqrt[3]{3 x}+2)(sqrt[3]{3 x}-2))
Answer

2.

  1. (71-22 sqrt{7})
  2. (sqrt[3]{x^{2}}-8 sqrt[3]{x}+15)

4.

  1. (27+8 sqrt{11})
  2. (29-12 sqrt{5})

6. (sqrt[3]{9 x^{2}}-4)

Divide Radical Expressions

Exercise (PageIndex{17}) Divide Square Roots

In the following exercises, simplify.

    1. (frac{sqrt{48}}{sqrt{75}})
    2. (frac{sqrt[3]{81}}{sqrt[3]{24}})
    1. (frac{sqrt{320 m n^{-5}}}{sqrt{45 m^{-7} n^{3}}})
    2. (frac{sqrt[3]{16 x^{4} y^{-2}}}{sqrt[3]{-54 x^{-2} y^{4}}})
Answer

2.

  1. (frac{8 m^{4}}{3 n^{4}})
  2. (-frac{x^{2}}{2 y^{2}})

Exercise (PageIndex{18}) rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

    1. (frac{8}{sqrt{3}})
    2. (sqrt{frac{7}{40}})
    3. (frac{8}{sqrt{2 y}})
    1. (frac{1}{sqrt[3]{11}})
    2. (sqrt[3]{frac{7}{54}})
    3. (frac{3}{sqrt[3]{3 x^{2}}})
    1. (frac{1}{sqrt[4]{4}})
    2. (sqrt[4]{frac{9}{32}})
    3. (frac{6}{sqrt[4]{9 x^{3}}})
Answer

2.

  1. (frac{sqrt[3]{121}}{11})
  2. (frac{sqrt[3]{28}}{6})
  3. (frac{sqrt[3]{9 x}}{x})

Exercise (PageIndex{19}) Rationalize a Two Term Denominator

In the following exercises, simplify.

  1. (frac{7}{2-sqrt{6}})
  2. (frac{sqrt{5}}{sqrt{n}-sqrt{7}})
  3. (frac{sqrt{x}+sqrt{8}}{sqrt{x}-sqrt{8}})
Answer

1. (-frac{7(2+sqrt{6})}{2})

3. (frac{(sqrt{x}+2 sqrt{2})^{2}}{x-8})

Solve Radical Equations

Exercise (PageIndex{20}) Solve Radical Equations

In the following exercises, solve.

  1. (sqrt{4 x-3}=7)
  2. (sqrt{5 x+1}=-3)
  3. (sqrt[3]{4 x-1}=3)
  4. (sqrt{u-3}+3=u)
  5. (sqrt[3]{4 x+5}-2=-5)
  6. ((8 x+5)^{frac{1}{3}}+2=-1)
  7. (sqrt{y+4}-y+2=0)
  8. (2 sqrt{8 r+1}-8=2)
Answer

2. no solution

4. (u=3, u=4)

6. (x=-4)

8. (r=3)

Exercise (PageIndex{21}) Solve Radical Equations with Two Radicals

In the following exercises, solve.

  1. (sqrt{10+2 c}=sqrt{4 c+16})
  2. (sqrt[3]{2 x^{2}+9 x-18}=sqrt[3]{x^{2}+3 x-2})
  3. (sqrt{r}+6=sqrt{r+8})
  4. (sqrt{x+1}-sqrt{x-2}=1)
Answer

2. (x=-8, x=2)

4. (x=3)

Exercise (PageIndex{22}) Use Radicals in Applications

In the following exercises, solve. Round approximations to one decimal place.

  1. Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of (75) square feet. Use the formula (s=sqrt{A}) to find the length of each side of his garden. Round your answers to th nearest tenth of a foot.
  2. Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was (175) feet. Use the formula (s=sqrt{24d}) to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.
Answer

2. (64.8) feet

Use Radicals in Functions

Exercise (PageIndex{23}) Evaluate a Radical Function

In the following exercises, evaluate each function.

  1. (g(x)=sqrt{6 x+1}), find
    1. (g(4))
    2. (g(8))
  2. (G(x)=sqrt{5 x-1}), find
    1. (G(5))
    2. (G(2))
  3. (h(x)=sqrt[3]{x^{2}-4}), find
    1. (h(-2))
    2. (h(6))
  4. For the function (g(x)=sqrt[4]{4-4 x}), find
    1. (g(1))
    2. (g(-3))
Answer

2.

  1. (G(5)=2 sqrt{6})
  2. (G(2)=3)

4.

  1. (g(1)=0)
  2. (g(-3)=2)

Exercise (PageIndex{24}) Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

  1. (g(x)=sqrt{2-3 x})
  2. (F(x)=sqrt{frac{x+3}{x-2}})
  3. (f(x)=sqrt[3]{4 x^{2}-16})
  4. (F(x)=sqrt[4]{10-7 x})
Answer

2. ((2, infty))

4. (left[frac{7}{10}, infty ight))

Exercise (PageIndex{25}) graph Radical Functions

In the following exercises,

  1. find the domain of the function
  2. graph the function
  3. use the graph to determine the range
  1. (g(x)=sqrt{x+4})
  2. (g(x)=2 sqrt{x})
  3. (f(x)=sqrt[3]{x-1})
  4. (f(x)=sqrt[3]{x}+3)
Answer

2.

  1. domain: ([0, infty))


  2. Figure 8.E.1
  3. range: ([0, infty))

4.

  1. domain: ((-infty, infty))


  2. Figure 8.E.2
  3. range: ((-infty, infty))

Use the Complex Number System

Exercise (PageIndex{26}) evaluate the Square Root of a Negative Number

In the following exercises, write each expression in terms of (i) and simplify if possible.

    1. (sqrt{-100})
    2. (sqrt{-13})
    3. (sqrt{-45})
Answer

Solve for yourself

Exercise (PageIndex{27}) Add or Subtract Complex Numbers

In the following exercises, add or subtract.

  1. (sqrt{-50}+sqrt{-18})
  2. ((8-i)+(6+3 i))
  3. ((6+i)-(-2-4 i))
  4. ((-7-sqrt{-50})-(-32-sqrt{-18}))
Answer

1. (8 sqrt{2} i)

3. (8+5 i)

Exercise (PageIndex{28}) Multiply Complex Numbers

In the following exercises, multiply.

  1. ((-2-5 i)(-4+3 i))
  2. (-6 i(-3-2 i))
  3. (sqrt{-4} cdot sqrt{-16})
  4. ((5-sqrt{-12})(-3+sqrt{-75}))
Answer

1. (23+14 i)

3. (-6)

Exercise (PageIndex{29}) Multiply Complex Numbers

In the following exercises, multiply using the Product of Binomial Squares Pattern.

  1. ((-2-3 i)^{2})
Answer

1. (-5-12 i)

Exercise (PageIndex{30}) Multiply Complex Numbers

In the following exercises, multiply using the Product of Complex Conjugates Pattern.

  1. ((9-2 i)(9+2 i))
Answer

Solve for yourself

Exercise (PageIndex{31}) divide Complex Numbers

In the following exercises, divide.

  1. (frac{2+i}{3-4 i})
  2. (frac{-4}{3-2 i})
Answer

1. (frac{2}{25}+frac{11}{25} i)

Exercise (PageIndex{32}) Simplify Powers of (i)

In the following exercises, simplify.

  1. (i^{48})
  2. (i^{255})
Answer

1. (1)

Practice Test

Exercise (PageIndex{33})

In the following exercises, simplify using absolute values as necessary.

  1. (sqrt[3]{125 x^{9}})
  2. (sqrt{169 x^{8} y^{6}})
  3. (sqrt[3]{72 x^{8} y^{4}})
  4. (sqrt{frac{45 x^{3} y^{4}}{180 x^{5} y^{2}}})
Answer

1. (5x^{3})

3. (2 x^{2} y sqrt[3]{9 x^{2} y})

Exercise (PageIndex{34})

In the following exercises, simplify. Assume all variables are positive.

    1. (216^{-frac{1}{4}})
    2. (-49^{frac{3}{2}})
  1. (sqrt{-45})
  2. (frac{x^{-frac{1}{4}} cdot x^{frac{5}{4}}}{x^{-frac{3}{4}}})
  3. (left(frac{8 x^{frac{2}{3}} y^{-frac{5}{2}}}{x^{-frac{7}{3}} y^{frac{1}{2}}} ight)^{frac{1}{3}})
  4. (sqrt{48 x^{5}}-sqrt{75 x^{5}})
  5. (sqrt{27 x^{2}}-4 x sqrt{12}+sqrt{108 x^{2}})
  6. (2 sqrt{12 x^{5}} cdot 3 sqrt{6 x^{3}})
  7. (sqrt[3]{4}(sqrt[3]{16}-sqrt[3]{6}))
  8. ((4-3 sqrt{3})(5+2 sqrt{3}))
  9. (frac{sqrt[3]{128}}{sqrt[3]{54}})
  10. (frac{sqrt{245 x y^{-4}}}{sqrt{45 x^{4} y^{3}}})
  11. (frac{1}{sqrt[3]{5}})
  12. (frac{3}{2+sqrt{3}})
  13. (sqrt{-4} cdot sqrt{-9})
  14. (-4 i(-2-3 i))
  15. (frac{4+i}{3-2 i})
  16. (i^{172})
Answer

1.

  1. (frac{1}{4})
  2. (-343)

3. (x^{frac{7}{4}})

5. (-x^{2} sqrt{3 x})

7. (36 x^{4} sqrt{2})

9. (2-7 sqrt{3})

11. (frac{7 x^{5}}{3 y^{7}})

13. (3(2-sqrt{3}))

15. (-12+8i)

17. (-i)

Exercise (PageIndex{35})

In the following exercises, solve.

  1. (sqrt{2 x+5}+8=6)
  2. (sqrt{x+5}+1=x)
  3. (sqrt[3]{2 x^{2}-6 x-23}=sqrt[3]{x^{2}-3 x+5})
Answer

2. (x=4)

Exercise (PageIndex{36})

In the following exercise,

  1. find the domain of the function
  2. graph the function
  3. use the graph to determine the range
  1. (g(x)=sqrt{x+2})
Answer

1.

  1. domain: ([-2, infty))


  2. Figure 8.E.3
  3. range: ([0, infty))

Solutions - Chapter 10

Open a blank file in your text editor and write a few lines summarizing what you’ve learned about Python so far. Start each line with the phrase In Python you can… Save the file as learning_python.txt in the same directory as your exercises fro mthis chapter. Write a program that reads the file and prints what you wrote three times. Print the contents once by reading in the entire file, once by looping over the file object, and once by storing the lines in a list and then working with them outside the with block.


PART 3: DISCIPLINARY ACTION (NON-PROBATIONARY REGULAR EMPLOYEES)

The supervisor may request action appropriate to the nature and severity of the offense or unacceptable performance and has the following options available. Items #2, #3, and #4 require prior approval from the Office of Employee and Labor Relations.

  1. Oral Reprimand or Warning: Supervisor prepares a memorandum of record for departmental file.
  2. Written Reprimand, Warning, or Notification of Unacceptable Performance: Supervisor prepares a memorandum to the employee, obtains approval from the Office of Employee and Labor Relations, and forwards a copy to the Office of Employee and Labor Relations for inclusion in the employee’s file.
  3. Suspension or Demotion: (See Part 7 below, Just Cause) Suspension of exempt employees will be for a minimum of 1 working day within a work week.
  4. Involuntary Termination: (See Part 7 below, Just Cause).

CBSE MASTER | NCERT Textbooks Exercises Solutions

Question 3. A electric circuit contains two UN-equal resistance in parallel
(a) Current is same in both
(b) More current flows through larger resistor
(c) Potential difference across each is same
(d) Smaller resistor has less conductance.

Answer. (c) Potential difference across each is same

Question 4. Ohm's law relates potential difference with ______?
(a) Power
(b) Energy
(c) Current
(d) Time

Question 5. Two bulbs marked 200 watt - 250 volts and 100 watt-250 Volts are joined in series to a 250 Volts supply. Power consumed in the circuit is ____?
(a) 33 Watts
(b) 67 Watts
(c) 100 Watts
(d) 300 Watts

Question 6. In what combination, the 3 resistors of 3 Ohms each should be connected to get effective resistance of 1 Ohm ?
(a) Series
(b) Parallel
(c) Any way
(d) None

Question 7. Two light bulbs are marked 230 V- 75 W and 230 V - 150 W, if the first bulb has resistance R, then the resistance of second is __ ?
(a) 4 R
(b) 2 R
(c) 1/4 R
(d) 1/2 R

Question 8. A copper wire has diameter of 0.5 mm and resistivity of 1.6 × 10 𕒼 Ω m. What will be the length of copper wire if the resistance of wire is 10 Ω
(a) 1.227m
(b) 12.27m
(c) 122.7m
(d) 0.1227m

Question 9. What should be the resistence of a Voltmeter ?
(a) Infinity
(b) Zero
(c) Can not say
(d) None of the above

Question 10. Three equal resistors, when combined in series gives an equivalent resistance of 90 Ω. What will be their equivallent resistance when combined in parallel ?
(a) 270 Ω
(b) 30 Ω
(c) 810 Ω
(d) 10 Ω

Question 11. A material which easily allows the flow of electrical charge through it, is called ___
(a) Insulator
(b) Conductor
(c) Semi Conductor
(d) Super Conductor

Question 12. Which property of electricity is responsible for use of fuse wire in household wiring ?
(a) Chemical effect
(b) Magnetic effect
(c) Heating effect
(d) All of above

Answer. (c) Heating effect

Question 13. A piece of wire of resistance R is cut into 4 equal parts. These parts are then connected in parallel. If the equivalent resistance of this combination is R′, then the ratio R/R′ is –
(a) 1/16
(b) 1/4
(c) 4
(d) 16

Question 14. When electric current is passed through a bulb, the bulb gives light because of
(a) Electric effect of current
(b) Magnetic effect of current
(c) Lighting effect of current
(d) Heating effect of current

Answer. (d) Heating effect of current

Question 15. Resistivity of a wire depends upon __
(a) Length of the wire
(b) Material of the wire
(c) Cross-section area of the wire
(d) All of the above

Answer. (d) All of the above

Question 16. The SI unit of resistivity is _
(a) Ω m
(b) Ω / m
(c) mho
(d) None

Question 17. The metals and alloys have very low resistivity, in the range of ___
(a) 10 𔃂 Ω m to 10 𔃀 Ω m
(b) 10 𔃄 Ω m to 10 𔃂 Ω m
(c) 10 𔃆 Ω m to 10 𔃄 Ω m
(d) 10 󈝶 Ω m to 10 𔃆 Ω m

Answer. (c) 10 𔃆 Ω m to 10 𔃄 Ω m

Question 18. Insulators like rubber and glass have very high resistivity, in the range of __
(a) 10 2 Ω m to 10 7 Ω m
(b) 10 4 Ω m to 10 11 Ω m
(c) 10 22 Ω m to 10 27 Ω m
(d) 10 12 Ω m to 10 17 Ω m

Answer. (d) 10 12 Ω m to 10 17 Ω m

Question 19. Two electric bulbs have resistance in the ratio of 1:2, if they are joined in series, the ratio of their energy consumption will be __
(a) 1:2
(b) 2:1
(c) 4:1
(d) 1:1

Question 20. A device used for measuring potential difference is known as ______
(a) Potentiometer
(b) Ammeter
(c) Voltmeter
(d) Galvanometer

True & False : Chapter 12. Electricity | CBSE Class 10th Science

Mark the below given statements as True or False

Question 1. Electric current can flow through metals

Question 2. Conductance is the property of a conductor to resist the flow of charges through it

Question 3. SI unit of electrical Potential Difference is Ampere (A)

Question 4. A voltmeter is used for measuring the electric current in the circuit.

Question 5. Resistance of conductor does not depend upon its length

Question 6. One volt is a potential difference between two points in current carrying conductor, when 1 joule of work is required to be done, in moving 1 coulomb of charge from one point to another.

Question 7. Electric cell is source of electrical energy
Answer. True

Question 8. The passage of electric current through a solution causes chemical reaction

Question 9. When current is passed through copper sulphate solution, the free copper gets accumulated on the electrode connected to positive terminal of the battery

Question 10. In an electrical circuit, potential difference (V) across the ends of given metallic wire, is inversely proportional to current flowing through it, provided its temperature remains the same. This is called Ohm's law.

Question 11. Materials which do not allow electric current to pass through them are called insulators

Question 12. In an electric circuit, the direction of the flow of current is from negative to positive terminal of the electrical cell.

Question 13. The filament of an electrical lamp is heated to such a high temperature that it starts glowing.

Question 14. The ammeter reading is reduced to approximately one half when length of the wire in the circuit is doubled.

Question 15. The process of depositing layer of any desired metal on another material by means of electricity is called electroplating.

Question 16. S.I. unit of electrical charge is Coulomb (C), which is equivalent to charge contained in nearly 6 吆 18 electrons

Question 17. The main disadvantage of a series circuit is that, when any one of the components breaks down, all other components stop working

Question 18. For the flow of charge in conducting metallic wire, the gravity has no role to play, the electrons move only if there is difference of electric pressure - called potential difference, along the conductor

Question 19. When electric current is passed through copper sulphate solution, the copper gets transferred from one electrode to other.

Question 20. Electrical current can pass through glass sheet.

Question 21. Pure water is good conductor of electricity

Question 21. Electrical Fuses are used in the circuit or appliances for lighting and heating purpose

Question 22. Conventionally, in an electrical circuit, the direction of electrical current is taken in the same direction as the flow of the electrons, which are negatively charged.

Question 23. Instead of a metallic wire a cotton thread can be used to make a circuit

Question 24. The metals and alloys have low electrical resistivity

Question 25. In order to maintain the electric current in the given circuit, the electric cell makes use of mechanical energy stored in it.

Question 26. ρ (rho) is a constant of proportionality and is called the electrical resistivity of the material of the conductor.

Question 27. The resistivity of an alloy is generally higher than that of its constituent metals

Question 28. Alloys do not oxidise (burn) readily at high temperatures. For this reason, they are commonly used in electrical heating devices, like electric iron, toasters etc

Question 29. The voltmeter is always connected in series across the points between which the potential difference is to be measured.

Question 30. The ammeter reading is increased when a thicker wire of the same material and of the same length is used in the circuit.

Question 31. When more than one cell is connected to each other, the combination is called a battery.

Question 32. when several resistors are joined in series, the resistance of the combination Rs equals the sum of their individual resistances

Question 33 Chromium plating is done on many object, because it has shiny appearance, it does not corrode, it resists scratches.

Question 34. Total potential difference across a combination of resistors in series is equal to the sum of potential difference across the individual resistors.
That is, V = V1 + V2 + V3

Question 35. A wire gets hot, when electric current is passed through it. This is due to magnetic effect of current

Solved Exercises : Chapter 12. Electricity | CBSE Class 10th Science

Question 1. A piece of wire of resistance R is cut into five equal parts. These parts are then connected in parallel. If the equivalent resistance of this combination is R′, then the ratio R/R′ is –
(a) 1/25
(b) 1/5
(c) 5
(d) 25

Question 2. Which of the following terms does not represent electrical power in a circuit?
(a) I 2 R
(b) IR 2
(c) VI
(d) V 2 /R

Question 3. An electric bulb is rated 220 V and 100 W. When it is operated on 110 V, the power consumed will be –
(a) 100 W
(b) 75 W
(c) 50 W
(d) 25 W

V 2
Electric Power P=
R
V 2
R=
P
220 𴢴
=
100
R=484 Ω
Now, When this bulb with Resistance 484 Ω is operated at 110 V
V 2
Electric Power P=
R
110 × 110

=
484
Power consumed by bulb=25 W

Question 4. Two conducting wires of the same material and of equal lengths and equal diameters are first connected in series and then parallel in a circuit across the same potential difference. The ratio of heat produced in series and parallel combinations would be –
(a) 1:2
(b) 2:1
(c) 1:4
(d) 4:1

Explanation :
Suppose the conducting wire has resistance R Ω
When ther wires are connected in series :

Question 5. How is a voltmeter connected in the circuit to measure the potential difference between two points?

Answer. Voltmeter is connected in parallel in the circuit to measure the potential difference between two points.

Question 6. A copper wire has diameter 0.5 mm and resistivity of 1.6 × 10𔃆 Ω m. What will be the length of this wire to make its resistance 10 Ω? How much does the resistance change if the diameter is doubled?

R=Resistance of conductor
ρ=Electrical resistivity of copper
l=Length of conductor
A=Area of cross-section of conductor
R=10 Ω
Diameter of wire=0.5 mm = 5/(10�) m
Radius of wire r=(5 × 10 𕒸 )/2
=2.5 × 10 𕒸 m
=25 × 10 𕒹 m

Hence the resistance of the wire will be reduced to 1/4 th of the earlier, if diameter of the wire is doubled

Question 7. The values of current I flowing in a given resistor for the corresponding values of potential difference V across the resistor are given below –


Plot a graph between V and I and calculate the resistance of that resistor.

Answer. Let us plot given values of Potential difference 'V' along Y-axis and that of current 'I' along X-axis.

While plotting a graph with given corresponding values of V and I, we have found that approximately the same value for V/I = 3.33 (approx.) is obtained in each case. Thus the V–I graph is a straight line that passes through the origin of the graph, as shown in Fig., V/I is a constant ratio.
A straight line plot shows that as the current through a wire increases, the potential difference across the wire increases linearly – this is Ohm’s law.

Question 8. When a 12 V battery is connected across an unknown resistor, there is a current of 2.5 mA in the circuit. Find the value of the resistance of the resistor.

Answer. As we know, according to Ohm's law

Question 9. A battery of 9 V is connected in series with resistors of 0.2 Ω, 0.3 Ω, 0.4 Ω , 0.5 Ω and 12 Ω, respectively. How much current would flow through the 12 Ω resistor?

Answer. When resistors are connected in series, the strength of current 'I' passing through each resistor is same.

V 9V 90
∴ I = = =
REq. 13.4 Ω 134
I = 0.67 Ampere
∴ value of current passing through 12 Ω resistor as well others = 0.67 Ampere.

Question 10. How many 176 Ω resistors (in parallel) are required to carry 5 A on a 220 V line?

Answer.
Let 'n' be the number of resistances, connected in parallel to 220 V to draw a maximum of 5 A current

1 1 1
= + +. n times
REq. 176 176
176
REq.= Ω
n

Now, according to Ohm's law :
V
R=
I

V
Or I=
R

220 × n
5 A=
176

5 𴢈 880
n= = =4
220 220
Hence, Number of Resistors, which can be connected in parallel across 220 V line to carry 5 A of current is 4

Question 11. Show how you would connect three resistors, each of resistance 6 Ω, so that the combination has a resistance of (i) 9 Ω, (ii) 4 Ω.

Answer. (i) To get a combination resistance of 9 Ω, we can connect the given three resistance of 6 Ω each as given below in the diagram :

As shown, a parallel combination of two resistance is connected in series with the third resistance.
R1×R2
REq= + R3
R1 + R2

6࡬
REq= + 6
6 + 6

36
REq= +6
12

REq=3+6=9 Ω
(ii) To get a combination resistance of 4 Ω, we can connect the given three resistance of 6 Ω each as given below in the diagram :


As shown, a series combination of two resistance is connected in parallel with the third resistance.
1 1 1
= +
REq R1 + R2 R3

(R1+ R2) × R3
REq=
(R1 + R2)+R3

(6+ 6) × 6 12࡬
REq= = =4 Ω
(6 + 6 ) + 6 18

Question 12. Several electric bulbs designed to be used on a 220 V electric supply line, are rated 10 W. How many lamps can be connected in parallel with each other across the two wires of 220 V line if the maximum allowable current is 5 A?

Voltage applied across the bulbs = 220 V
Maximum allowable current = 5 A
Power rating of each bulb W = 10 W
Let R be resistance of each bulb

V
R=
I
P
I=
V
V 2
∴ R=
P
220 × 220
∴ R=
10
∴ R=4840 Ω

Let 'n' be the number of bulbs, connected in parallel to 220 V to draw a maximum of 5 A current
1 1 1
= + +. n times
REq. 4840 4840

4840
REq.= Ω
n

Now, according to Ohm's law :
V
R=
I

V
Or I=
R

220 × n
5 A=
4840

5 � 24200
n= = =110
220 220
Hence, Number of bulbs which can be connected in parallel with each other across the two wires of 220 V line when the maximum allowable current is 5 A = 110

Alternate Solution: !!
Vlotage applied across the bulbs = 220 V
Maximum allowable current = 5 A
Power rating of each bulb W = 10 W
Max. Power capacity of the line , P = VI
P= 220 × 5 = 1100 W
let 'n' be the Numbers of bulbs of 10 Watt each which can be connected in parallel
Therefor n= Total Power Capacity of line÷Power Rating of each bulb
n= 1100悆 = 110
Hence 110 bulbs can be connected in parallel across the line

Question 13. A hot plate of an electric oven connected to a 220 V line has two resistance coils A and B, each of 24 Ω resistance, which may be used separately, in series, or in parallel. What are the currents in the three cases?

Case 1. When electric oven is connected with just one resistance coil

V 220
I1= =9.2 A
R 24

Case 2. When electric oven is connected with two resistance coils in series
∴ R = 24 + 24 = 48 Ω

V 220
I2= = =4.6 A approx.
R 48

Case 3. When electric oven is connected with two resistance coils in parallel
24 × 24 24 × 24
R= = =12 Ω approx.
24 + 24 48
V 220
I3= = =18.3 A approx.
R 12

Question 14. Compare the power used in the 2 Ω resistor in each of the following circuits: (i) a 6 V battery in series with 1 Ω and 2 Ω resistors, and (ii) a 4 V battery in parallel with 12 Ω and 2 Ω resistors.


V=6 Volt
R=1 + 2 = 3 Ω
V 6
I= =
R 3
I=2 A
Let P1 be power used in 2 Ω resistor
P1=VI=6ࡨ = 12 W
Case 2. : When resistor is connected in parallel :


V=4 Volt
R1×R2
R=
R1+R2
12ࡨ
R=
12+2
R=1.71 Ω
V 4
I= =
R 1.7
I=2.34 A
Let P2 be power used in 2 Ω resistor
P2=VI=4ࡨ.34= 9.36 W
Comparison of power used : Ratio
P1:P1::12 : 9.36
P1:P1::1 : 0.78

Question 15. Two lamps, one rated 100 W at 220 V, and the other 60 W at 220 V, are connected in parallel to electric mains supply. What current is drawn from the line if the supply voltage is 220 V?

Power rating of first lamp=100 W
Power rating of second lamp=60 W
Voltage applied across lamps=220 V
Let, Resistance of first lamp=R1
Resistance of second lamp=R2
V 2
As we know, R=
P
220 𴢴
R1= = 480 Ω
100
220 𴢴
R2= = 806.66 Ω
60
When R1 and R2 are connected in parallel, Let be REq equivalent resistance and I be current drawn through R1 and R2 connected in parallel.
1 1 1
= +
REq R1 R2
R1×R2
REq=
R1+R2
484𴫾.66
REq= Ω
484+806.66
390423.44
REq= = 302.5 Ω
1290.66
V 220
I= =
R 302.5
I=0.727 A

Question 16. Which uses more energy, a 250 W TV set in 1 hr, or a 1200 W toaster in 10 minutes?

For TV set :
E1=250 W × 1 × 3600 sec
=900000 J = 9 × 10 5 J
For Toaster :
E2=1200 W × 10 × 60 sec
=720000 J = 7.2 × 10 5 J
=>E1>E2
∴ TV Set uses more power.

Question 17. An electric heater of resistance 8 Ω draws 15 A from the service mains 2 hours. Calculate the rate at which heat is developed in the heater.

Heat energy developed in the heater= I 2 Rt
Rate of Heat Energy= Power
P= I 2 Rt / t
P= I 2 R
P = 15 2 × 8
P = 1800 Watt

∴ Heat is developed at the rate of 1800 W or 1.8 KW or 1800 joule per sec.

Question 18. Explain the following.
(a) Why is the tungsten used almost exclusively for filament of electric lamps?
(b) Why are the conductors of electric heating devices, such as bread-toasters and electric irons, made of an alloy rather than a pure metal?
(c) Why is the series arrangement not used for domestic circuits?
(d) How does the resistance of a wire vary with its area of cross-section?
(e) Why are copper and aluminum wires usually employed for electricity transmission?

Answer.
(a) Tungsten offers high resistance against electric current and it has high melting point. Mainly for these two characteristics, Tungsten is used almost exclusively for filament of electric lamps. When the electric current flows through Tungsten filament, due to high resistance of Tungsten, it gets heated up to a very high temperature and starts glowing with light. High melting point of Tungsten prevents it from melting away at higher temperature.

  1. It offers high resistance against an electric current
  2. It can withstand high temperature due to high melting poin.
  3. It has high density

The that heat produced in a resistor is (i) directly proportional to the square of current for a given resistance
(ii) directly proportional to resistance for a given current, and (iii) directly proportional to the time for which the current flows through the resistor.
Higher resistance, high melting point and higher density of an alloy in conductors of electric heating devices, all to gather, help in producing a large amount of heat at higher temperature, at the same time offsetting any deformation arising out of temperature variations.

(c) When appliances are connected in series, the value of equivalent resistance will be very large, as shown below :-

This may result in either loss of electric energy due to heating or melting away of electric wiring in domestic circuit due to over heating. Any fault in a series arrangement results in complete shutdown of other devices in circuit. When appliances are connected in parallel, the value of equivalent resistance will be very small hence damage or energy losses due to heating will be minimum. Also break down of a certain device does not affect others in the parallel circuit
This is the main reason, why the series arrangement is not used for domestic circuits

R=Resistance of conductor
ρ=Electrical resistivity of copper
l=Length of conductor
A=Area of cross-section of conductor

From the above, it is clear that as the radius of cross sectional area of a conductor increases, the resistance of conductor decreases and vice-versa

(e) Certain elements like Silver, Copper and Aluminum are the best conductors of electricity due to theirs low electric resistivity. Hence, voltage drop and heating loss due to resistance against electric current is minimum. Silver being too costly, Copper and Aluminum wires are usually employed for electricity transmission

Intext Questions|Page 200 |Chapter 12. Electricity | CBSE Class 10th Science

Question 1.What does an electric circuit mean?

Answer : An electrical circuit essentially consists of load devices, connected together by conducting material such as wire and electric power source. The naure of load devices, depending upon the application, may be either Resistive or Inductive or Capacitive .

Question 2. Define the unit of current.

Answer : The rate of flow of electric charges, flowing through a particular area of a conductor is called Electric current. S.I. unit of Electric current is ampere (A). One ampere of current represents a flow of one coulomb of charge, passing through a particular area of a conductor, per second. Coulomb is the SI unit of electric charge and one coulomb (C) of charge is equivalent to the charge contained in nearly 6 × 10 18 electrons

Question 3.Calculate the number of electrons constituting one coulomb of charge.

We know that an electron possesses a negative charge of 1.6 × 10 󈝿 C
Let 'n' be the number of electrons present in one coulomb of charge

Number of electrons in Coulomb × Charge per Electron = One coulomb of charge


1 10 19 100吆 18
n= = =
1.6 × 10 󈝿 1.6 16

n = 6.25 × 10 18 electrons

Intext Questions|Page 202 |Chapter 12. Electricity | CBSE Class 10th Science

Question 1. Name a device that helps to maintain a potential difference across a conductor.

Answer : Electric cell is the device that helps to maintain a potential difference across a conductor. The chemical action within a cell generates the potential difference across the terminals of the cell, even when no current is drawn from it. When the cell is connected to a conducting circuit element, the potential difference sets the charges in motion in the conductor and produces an electric current

Question 2. What is meant by saying that the potential difference between two points is 1 V?

Answer : One volt is the potential difference between two points in a current carrying conductor when 1 joule of work is done to move a charge of 1 coulomb from one point to the other. Volt (V) is the SI unit of electric potential difference

Question 3. How much energy is given to each coulomb of charge passing through a 6 V battery?

Answer : We define the electric potential difference (V) between two points in an electric circuit carrying some current as the work (W) done or Energy in joule to move a unit charge(Q) from one point to the other –

Potential difference (V) between two points = Work done (W)/Charge (Q)

Intext Questions|Page 209 |Chapter 12. Electricity | CBSE Class 10th Science

Question 1. On what factors does the resistance of a conductor depend?

Answer : Resistance of the conductor depends on following factors :

(i) Length of conductor (l)
: Resistance of a uniform metallic conductor is directly proportional to its length

(ii) The Cross-section area of the conductor (A) :
Resistance of a uniform metallic conductor is inversely proportional to the area of cross-section (A).

Question 2. Will current flow more easily through a thick wire or a thin wire of the same material, when connected to the same source? Why?

The current is inversely proportional to Resistance and further Resistance of a uniform metallic conductor is inversely proportional to the area of cross-section (A). Which means more the area of cross section, lesser will be the resistance and therefor more will be the current flow. Hence, current Will flow more easily through a thick wire than a thin wire of the same material, when connected to the same source

Question 3. Let the resistance of an electrical component remains constant while the potential difference across the two ends of the component decreases to half of its former value. What change will occur in the current through it?

V (Potential difference) = Current (I) × Resistance (R)

As the Resistance (R) is constant.

When V (Potential difference) decreases to half of its former value

Question 4.Why are coils of electric toasters and electric irons made of an alloy rather than a pure metal?

Answer : The resistivity of an alloy is generally higher than that of its constituent pure metals. Higher resistance results in increased electrical heating. Alloys do not oxidise (burn) easily at high temperatures. For this reason, they are commonly used in electrical heating devices, like electric iron, toasters etc

Question 5. Use the data in Table 12.2 to answer the following –
(a) Which among iron and mercury is a better conductor?
(b) Which material is the best conductor?

Answer : (a) Iron is a better conductor than mercury as its Electrical resistivity is less (10.0 × 10 𔃆 Ω m), whereas that of mercury is very high.( 94.0 × 10 𔃆 Ω m)

(b) Silver is the best conductor of electric current because of its Electrical resistivity is 1.60 × 10𔃆 Ω m, which is least of all other metals.


CBSE MASTER | NCERT Textbooks Exercises Solutions

Question 2. The image formed by a concave mirror is observed to be virtual, erect and larger than the object. Where should be the position of the object?
(a) Between the principal focus and the centre of curvature
(b) At the centre of curvature
(c) Beyond the centre of curvature
(d) Between the pole of the mirror and its principal focus.

Answer. (d) Between the pole of the mirror and its principal focus.

Question 3. Where should an object be placed in front of a convex lens to get a real image of the size of the object?
(a) At the principal focus of the lens
(b) At twice the focal length
(c) At infinity
(d) Between the optical centre of the lens and its principal focus.

Answer. (c) At infinity

Question 4. A spherical mirror and a thin spherical lens have each a focal length of 󈝻 cm. The mirror and the lens are likely to be
(a) both concave.
(b) both convex.
(c) the mirror is concave and the lens is convex.
(d) the mirror is convex, but the lens is concave.


Answer. (a) both concave.

Question 5. No matter how far you stand from a mirror, your image appears erect. The mirror is likely to be
(a) plane.
(b) concave.
(c) convex.
(d) either plane or convex.

Answer. (d) either plane or convex.

Question 6. Which of the following lenses would you prefer to use while reading small letters found in a dictionary?
(a) A convex lens of focal length 50 cm.
(b) A concave lens of focal length 50 cm.
(c) A convex lens of focal length 5 cm.
(d) A concave lens of focal length 5 cm.

Answer. (c) A convex lens of focal length 5 cm.

Question 7. We wish to obtain an erect image of an object, using a concave mirror of focal length 15 cm. What should be the range of distance of the object from the mirror? What is the nature of the image? Is the image larger or smaller than the object? Draw a ray diagram to show the image formation in this case.

Answer. (a). The range of distance of the object from the mirror should be less than 15 cm i.e. from 0 to 15 cm in the front of mirror from the pole.
(b). The nature of image so formed will be virtual and erect.
(c) The size of image will be larger than object

Question 8. Name the type of mirror used in the following situations.
(a) Headlights of a car.
(b) Side/rear-view mirror of a vehicle.
(c) Solar furnace.
Support your answer with reason.

Answer. (a) In headlights of a car, type of mirror used is concave as the light of the lamp, under goes divergence from reflector surface and cover a large area in the front.
(b) Side/rear-view mirror of a vehicle is a convex mirror as it gives a diminished, Virtual and an erect image of the side or rear with wider field of view . A convex mirrors enable the driver to view much larger area than would be possible with a plane mirror
(c) Solar furnace is a concave mirror as Sun rays after reflection from its surface, get converged at focus with much intense heat.

Question 9. One-half of a convex lens is covered with a black paper. Will this lens produce a complete image of the object? Verify your answer experimentally. Explain your observations.

Answer. Yes, the half covered lens will still produce a complete image of the object but the image so formed may not be as intense as formed with uncovered lens. In fact, parts or broken pieces of lens behave like a complete lens and form complete image.
Verification : Take a convex lens. Light a candle. Now form an image of a burning candle on a white surface on the other side of lens by adjusting the distance between lens and candle. We can observe, a complete real and an inverted image of candle. Now cover half of lens with black paper, and try to form an image now. We can observe that again a complete, real and an inverted image of candle is formed. Image formed was less intense from earlier.

Question 10. An object 5 cm in length is held 25 cm away from a converging lens of focal length 10 cm. Draw the ray diagram and find the position, size and the nature of the image formed.

Answer. Height of the object h = + 5 cm
Focal length f = + 10 cm
object-distance u = 󈞅 cm
Image-distance v = ?
Height of the image h′ = ?


1
1
1
=
v
u
f
=>1
1
1
=
v
10
25
=>
5 − 2
3
= =

50
50
=>
50
v= =16.66 cm

3
A real and inverted image will be formed on other side of lense at 16.66 cm from its optical centre
=>
v
m=

u
=>h2
16.66
=
h1
− 25
=>h2
16.66
=
5
− 25
=>
󔼘.66 × 5
h2=

25
=>

h2=− 3.33 cm



An inverted, 3.33 cm high image will be formed.


Question 11. A concave lens of focal length 15 cm forms an image 10 cm from the lens. How far is the object placed from the lens? Draw the ray diagram.

Answer. Focal length f = − 15 cm
Image-distance v = − 10 cm
object-distance u = ?


1
1
1
=
v
u
f

1
1
1
=
− 10
u
− 15

− 1
1
1
+ =
10
15
u
=>1 − 3 + 2
=
u
30
=>u= − 30 cm

Object is placed at a distance of 30 cm from concave lens

Question 12. An object is placed at a distance of 10 cm from a convex mirror of focal length 15 cm. Find the position and nature of the image.

Answer. Focal length f = 15 cm
object-distance u = 󈝶 cm
Image-distance v = ?

1 1 1
+ =
v u f
1 1 1
+ =
v − 10 15
1 1 1
= +
v 15 10
1 2+3 5
= =
v 30 30
v= 6 cm

A virtual and erect image is formed 6 cm behind the mirror.

Question 13. The magnification produced by a plane mirror is +1. What does this mean?

Answer. The magnification produced by a plane mirror is +1 implies that the image formed by a plane mirror is virtual , erect and of the same size, as that of object.

Question 14. An object 5.0 cm in length is placed at a distance of 20 cm in front of a convex mirror of radius of curvature 30 cm. Find the position of the image, its nature and size.

Answer.
Radius of curvature, R = + 30 cm
Focal length, f = R/2 = + 30/2 cm = + 15 cm
Object-distance, u = – 20 cm
Height of object
h1′= 5 cm
Image-distance, v = ? Height of image h2′= ?


1
1
1
+ =
v
u
f

1
1
1
+ =
v
− 20
+ 15

1
1
1
= +
v
15
20

1
4 + 3 7
= =
v
60
60


60
v= = 8.57 cm

7
The image is formed behind the mirror at a distance of 8.6 cm


h2 − v
m= =

h1 u


h2 8.57
=>= =

5 cm 20


8.57 × 5 cm
= h2=

20


Height (Size ) of Image= h2= 2.175 cm


Thus, a 2.175 cm high, virtual and erect image is formed

Question 15. An object of size 7.0 cm is placed at 27 cm in front of a concave mirror of focal length 18 cm. At what distance from the mirror should a screen be placed, so that a sharp focussed image can be obtained? Find the size and the nature of the image.

Answer. Focal length, f = - 18 cm
Object-distance, u = – 27 cm
Height of object, h1′= 5 cm
Image-distance, v = ?
Height of image h2′= ?


1
1
1
+ =
v
u
f

1
1
1
+ =
v
− 27
-18 15

− 1
1
1
= +
v
18
27

1
− 3 + 2 1
= =
v
54
54


60
v= = 54 cm

7
The screen should be kept at a distance of 54 cm in front of mirror


h2 − v
m= =

h1 u


h2 (− 54 )
=>= =

7 cm (− 27 )


(− 54 ) × 7 cm
=> h2=

(− 27 )


Height (Size ) of Image= h2= 2 × 7 cm= 14 cm cm


Thus, a 14 cm high, virtual and an inverted image is formed

Question 16. Find the focal length of a lens of power – 2.0 D. What type of lens is this?

Answer.




1
P=


f



1
- 2.0 =


f



𕒵
f = m


2



− 1
f= × 100 cm


2



f =− 50cm = - 0.50 cm
The lens is a concave lens

Question 17. A doctor has prescribed a corrective lens of power +1.5 D. Find the focal length of the lens. Is the prescribed lens diverging or converging?

Answer. Power of lens, P = + 1.5 D




1
P=
f



1
1.5=
f



1
f= m
1.5



10
f= m
15



10
f= m
15



2
f= m = + 0.67 m
3

Focal length of the lens is + 0.67 m. The prescribed lens is converging type in nature.


Additional Questions | Chapter 10. Light – Reflection and Refraction | CBSE Class 10th Science


Question 1. Write the mirror formula :

Answer. In mirror, the distance of the object from its pole is called the object distance (u)
The distance of the image from the pole of the mirror is called the image distance (v)
The distance of the principal focus from the pole is called the focal length (f).
There is a relationship between these three quantities given by the mirror formula which is expressed as :

1 1 1
+ =
v u f

Question 2. Write one use of concave and convex mirrors each.

Answer. Concave mirror is used in some telescopes and also as magifying looking mirror while applying make-up or shaving. Convex mirror is used in vehicles as rear view mirror.

Question 3. Why do we use convex for side-view ?

Answer. Convex mirrors are used in vehicles for side-view because they give a virtual, upright, though diminished, image. Although images so formed are smaller but this results in showing a large area in the back drop with a wider field of view .

Question 4. How does a light ray bend when it travels from :
(i) A denser to rarer medium ?
(ii) Ararer to denser medium >

Answer.An optically denser medium has a larger refractive index, where as optically rarer medium has a lower refractive index. Due to refration, the speed of light is higher in a rarer medium than a denser medium also direction of propagation of light chages as it chages the medium. . (i) When light travels from a denser medium to a rarer medium, it speeds up and bends away from the normal.
(ii) When light travels from a rarer medium to a denser medium, it slows down and bends towards the normal.

Question 5. When a convex lens is focussed on a distant object, where will the image be formed ? Show it with a ray diagram.

Answer. When a convex lens is focussed on a distant object, the image will be formed at the focus of the lens.

Question 6. What is the meaning of
(i) Optical Centre
(ii) Principal Axis

Answer. (i) Optical Centre : The central point of a lens is its optical centre. It is usually represented by the letter O. A ray of light through the optical centre of a lens passes without suffering any deviation.
(ii) Principal Axis : An imaginary straight line passing through the two centres of curvature of a lens is called its principal axis. Optical centre and focus of lens lies on the Principal Axis.

Question 7. When does a convex lens form
(i) A virtual, erect, enlarged image ?
(ii) A real enlarged image ?

Answer. (i) A convex lens forms a virtual, erect and enlarged image, when the object is placed between focus and optical centre of a lens on its other side.
(ii) A convex lens forms a real and enlarged image when the object is placed between focus (f) and centre of curvature (2f) of a lens on its other side.

Question 8. What is the relationship between the focal length of a spherical mirror and radius of curvature ?

Answer. The focal length of a spherical mirror (f) is equal to half its radius of curvature (R) i.e.
Focal length, f = R/2

Question 9. Explain the term Magnification produced by a spherical mirror ?

Answer. Magnification produced by a spherical mirror gives the relative extent to which the image of an object is magnified with respect to the object size. It is expressed as the ratio of the height of the image to the height of the object. It is usually represented by the letter m.
If h1 is the height of the object and h2 is the height of the image, then the magnification m produced by a spherical mirror is given by

Question 10. Name and explain the sign Convention for Reflection by Spherical Mirrors

Answer. While dealing with the reflection of light by spherical mirrors, we follow a set of sign conventions called the New Cartesian Sign Convention. In this convention, the pole (P) of the mirror is taken as the origin. The principal axis of the mirror is taken as the x-axis (X’X) of the coordinate system. The conventions are as follows –
(i) The object is always placed to the left of the mirror. This implies that the light from the object falls on the mirror from the left-hand side.
(ii) All distances parallel to the principal axis are measured from the pole of the mirror.
(iii) All the distances measured to the right of the origin (along + x-axis) are taken as positive while those measured to the left of the origin (along – x-axis) are taken as negative.
(iv) Distances measured perpendicular to and above the principal axis (along + y-axis) are taken as positive.
(v) Distances measured perpendicular to and below the principal axis (along –y-axis) are taken as negative.

Question 11. Explain the following given terms
(i) Refraction of light
(ii) Laws of refraction of light
(ii) The Refractive Index

Answer.
(i) Refraction of light : The direction of propagation of light, When traveling obliquely from one medium to another is subject to change. When light travels from a denser medium to a rarer medium, it speeds up and bends away from the normal. When light travels from a rarer medium to a denser medium, it slows down and bends towards the normal. This phenomenon of bending of light ray is known as refraction of light. Refraction is caused due to change in the speed of light as it enters from one transparent medium to another. The speed of light increases in rarer medium and decreases in denser medium.

(ii) Laws of refraction of light : The reflecting surfaces, of all types, obey the laws of reflection. The refracting surfaces obey the laws of refraction.
The following are the laws of refraction of light.
(i) The incident ray, the refracted ray and the normal to the interface of two transparent media at the point of incidence, all lie in the same plane.
(ii) The ratio of sine of angle of incidence to the sine of angle of refraction is a constant, for the light of a given colour and for the given pair of media. This law is also known as Snell’s law of refraction.
If i is the angle of incidence and r is the angle of refraction, then,


Intext Questions | Page 168 | Chapter 10. Light – Reflection and Refraction | CBSE Class 10th Science

Question 1. Define the principal focus of a concave mirror.

Answer. The principal focus of the concave mirror is an point on the principal axis of the mirror, where incident rays of light, parallel to the principal axis, after reflection from mirror surface, intersect each other .

Question 2. The radius of curvature of a spherical mirror is 20 cm. What is its focal length?

Answer. For spherical mirrors of small apertures, the radius of curvature is twice the focal length i.e. R = 2f . Here radius of curvature R = 20 cm, focal length f = ?




R
20
f= = = 10 cm


2
2

Focal length of mirror is 10 cm

Question 3. Name a mirror that can give an erect and enlarged image of an object.

Answer. Concave mirror gives an erect and enlarged image of an object, when the object is between pole (P) and principal focus of mirror (C).

Question 4. Why do we prefer a convex mirror as a rear-view mirror in vehicles?

Answer. Convex mirrors are commonly used as rear-view (wing) mirrors in vehicles because they give an erect, virtual, full size diminished image of distant objects with a wider field of view. Thus, convex mirrors enable the driver to view much larger area than would be possible with a plane mirror.


Intext Questions | Page 171 | Chapter 10. Light – Reflection and Refraction | CBSE Class 10th Science

Question 1. Find the focal length of a convex mirror whose radius of curvature is 32 cm.

Answer.We know,for spherical mirrors of small apertures, the radius of curvature is twice the focal length i.e. R = 2f .
Given here, radius of curvature R is 32 cm. and focal length of a convex mirror f = ?




R
32
f= = = 16 cm


2
2

Focal length of mirror is 16 cm

Question 2. A concave mirror produces three times magnified (enlarged) real image of an object placed at 10 cm in front of it. Where is the image located?

Answer. Here given Magnification m = 3, Object-distance u = 10 cm





v
Magnification m= Real image



u




v
− 3 =



u






3 u =v












v =3u= 3 × − 10 cm = − 30 cm






The image will be formed at a distance of 30 cm in front of convex mirror from its the pole

Intext Questions | Page 176 | Chapter 10. Light – Reflection and Refraction | CBSE Class 10th Science

Question 1. A ray of light travelling in air enters obliquely into water. Does the light ray bend towards the normal or away from the normal? Why?

Answer. The light ray bends towards the normal as it travels from a rarer medium of air to a denser medium of water, under goes refraction. Refraction is due to change in the speed of light as it enters from one transparent medium to another. The speed of light increases in rarer medium and decreases in denser medium

Question 2. Light enters from air to glass having refractive index 1.50. What is the speed of light in the glass? The speed of light in vacuum is 3 × 10 8 m s 𔂿 .

Answer. Given, speed of light in vacuum C = 3 × 10 8 m s 𔂿
Refractive index of glass ng = 1.50
Speed of light in the glass vg = ?




C
ng=


Vg



3 × 10 8
ng=


Vg



3 × 10 8
1.5=


Vg



3 × 10 8
Vg=


1.5




Vg=2 × 10 8 ms 𕒵




Speed of light in the glass vg = 2 × 10 8 ms 𕒵

Question 3. Find out, from Table 10.3, the medium having highest optical density. Also find the medium with lowest optical density.

Answer.The medium having highest optical density : Diamond ( Refractive Index 2.42 )
The medium having lowest optical density : Air ( Refractive Index 1.0003 )

Question 4. You are given kerosene, turpentine and water. In which of these does the light travel fastest? Use the information given in Table 10.3.

Answer. Using the information given in table, the refrative index of kerosene is 1.44, that of turpentine is 1.47 and that of water is 1.33. Clearly, water having lower refractive index 1.33, is optically rarer than kerosene and turpentine. Therefor the light travels fastest in water because of its lower optical density

Question 5. The refractive index of diamond is 2.42. What is the meaning of this statement?

Answer. The diamond with refractive index of 2.42, is the most ‘optically denser medium' and there for the speed of light in diamond will be less i.e. 1.23 × 10 8 ms 𕒵 (= 3 × 10 8 ms 𕒵 /2.42 ) compare to speed of light in vacuum C = 3 × 10 8 m s - 1

Intext Questions | Page 184 | Chapter 10. Light – Reflection and Refraction | CBSE Class 10th Science

Question 1. Define 1 dioptre of power of a lens.

Answer. 1 dioptre is SI unit of the power of a lens whose focal length is 1 metre. It is denoted by the letter D. Thus 1D = 1 m -1 . Simply, when focal length ' f ' is expressed in metres, then, power is expressed in dioptres. The power of a convex lens is positive and that of a concave lens is negative.

Question 2. A convex lens forms a real and inverted image of a needle at a distance of 50 cm from it. Where is the needle placed in front of the convex lens if the image is equal to the size of the object? Also, find the power of the lens.

Answer.


v= 50
m= 1



1
P= = ?

f

As the image formed by lens is real




v
m=

u


v
1=

u



v= u



u= 50 cm

∴ Object distance is = 50 cm

1
1
1
=
v
u
f

1
1
1
=
50
− 50
f

1 + 1
1
=
50
f

2 f =50

f =25 cm = .25 m



1
100

P= + = + =+ 4 dioptre

.25 25


The power of convex lens is + 4 dioptre

Question 3. Find the power of a concave lens of focal length 2 m.


Focal length of lens f=− 2 m
Power of concave lens P=?





1
P=



f




1
P=



2


P=0.5 dioptre

The power of concave lens is − 0.5 dioptre

  • Take a large shining spoon. Try to view your face in its curved surface.
  • Q.1. Do you get the image? Is it smaller or larger?
  • Answer : Yes, the image of the face formed on outer curved surface is smaller in size.
  • Q.2. Move the spoon slowly away from your face. Observe the image. How does it change?
  • Answer : The size of image gradually decreases with a increase in field of view.
  • Q.3. Reverse the spoon and repeat the Activity. How does the image look like now?
  • Answer : Earlier, when the spoon was close the image formed on the inner curved surface was erect and magnified and as we moved the spoon slowly away from our face,the image transitioned to a inverted image with gradual decrease in its size.
  • Q.4. Compare the characteristics of the image on the two surfaces.
  • Outer SurfaceInner Surface
    (i) Image is always erect
    (ii) Image size is gradually decreases as we move away the spoon
    (i) The image is erect when spoon is close and inverted when spoon is away
    (ii) Image size is larger when spoon is close and it is smaller when spoon is moved away
  • Q. 5. Why do we see our image in the shining spoon?
  • Answer : The surface of a shining spoon acts like a mirror. Due to reflection of light from its surfaces, we can see our image
  • Q. 6. What types of mirrors are formed by the inner and outer curved surfaces of a spoon ?
  • Answer : The inner curved surfaces of a spoon forms a concave mirror and the outer curved surfaces of a spoon forms a convex mirror
  • Hold a concave mirror in your hand and direct its reflecting surface towards the Sun.
  • Direct the light reflected by the mirror on to a sheet of paper held close to the mirror.
  • Move the sheet of paper back and forth gradually until you find on the paper sheet a bright, sharp spot of light.
  • Q.1. Hold the mirror and the paper in the same position for a few minutes. What do you observe? Why?
  • Answer : The light rays from the Sun, are concentrated at focus, and form a sharp spot of light on the sheet. Paper starts burning after some time due to increase intensity of reflected sun light from the mirror
  • Q.2. Why, we should not look at the Sun directly or even into a mirror reflecting sunlight
  • Answer : Because the intense heat resulting from the concentrated sun light through eye lens may burn the retina wall with dark spots. This may result in partial or complete vision impairment.
  • Take a concave mirror. Find out its approximate focal length in the way described above. Note down the value of focal length. (You can also find it out by obtaining image of a distant object on a sheet of paper.)
  • Mark a line on a Table with a chalk. Place the concave mirror on a stand. Place the stand over the line such that its pole lies over the line.
  • Draw with a chalk two more lines parallel to the previous line such that the distance between any two successive lines is equal to the focal length of the mirror. These lines will now correspond to the positions of the points P, F and C, respectively. Remember – For a spherical mirror of small aperture, the principal focus F lies mid-way between the pole P and the centre of curvature C.
  • Keep a bright object, say a burning candle, at a position far beyond C. Place a paper screen and move it in front of the mirror till you obtain a sharp bright image of the candle flame on it.
  • Observe the image carefully. Note down its nature, position and relative size with respect to the object size.
  • Repeat the activity by placing the candle – (a) just beyond C, (b) at C, (c) between F and C, (d) at F, and (e) between P and F.
  • In one of the cases, you may not get the image on the screen. Identify the position of the object in such a case. Then, look for its virtual image in the mirror itself.
  • Q. 1. Note down and tabulate your observations.
  • Answer : Image formation by a concave mirror for different positions of the object

  • Q. 1. Draw neat ray diagrams for each position of the object shown in Table 10.1.
  • Answer :

  • You may take any two of the rays mentioned in the previous section for locating the image.
  • Compare your diagram with those given in Fig. 10.7 of textbook.
  • Answer : They were identical and matching
  • Describe the nature, position and relative size of the image formed in each case.
  • Tabulate the results in a convenient format.
  • Take a convex mirror. Hold it in one hand.
  • Hold a pencil in the upright position in the other hand.
  • Q. 1. Observe the image of the pencil in the mirror. Is the image erect or inverted? Is it diminished or enlarged?
  • Answer : The image is erect and diminished
  • Q. 2. Move the pencil away from the mirror slowly. Does the image become smaller or larger?
  • Answer : The image becomes smaller.
  • Repeat this Activity carefully. State whether the image will move closer to or farther away from the focus as the object is moved away from the mirror?
  • Answer : The image will move closer to the focus
  • Observe the image of a distant object, say a distant tree, in a plane mirror.
  • Q. 1. Could you see a full-length image?
  • Answer : No, We can not see a full-length image of a distant object in a plain mirror.
  • Q. 1. Try with plane mirrors of different sizes. Did you see the entire object in the image?
  • Answer : No, the result was same as before.
  • Q. 4. Repeat this Activity with a concave mirror. Did the mirror show full length image of the object?
  • Answer : No
  • Q. 4. Now try using a convex mirror. Did you succeed? Explain your observations with reason.
  • Answer : Yes, now we could see full length image of distant object with wider field of view. The image formed was diminished, erect and virtual. Due to this reason, they are used as rear or side view mirrors in vehicles as distant objects in the backdrop can be seen clearly with much larger field of view .
  • Place a coin at the bottom of a bucket filled with water.
  • Q. 1. With your eye to a side above water, try to pick up the coin in one go. Did you succeed in picking up the coin?
  • Answer : No
  • Q. 2. Repeat the Activity. Why did you not succeed in doing it in one go?
  • Answer : Because on seeing, the coin appeared to be closer than its actual distance, so we are likely to miss the coin. Reflected light coming from the submerged coin in denser medium of water, on entering air which is a rarer medium, bend away from the normal due to refraction of light and image size becomes larger than its actual size, thus submerged object apparently seem closer.
  • Ask your friends to do this. Compare your experience with theirs.
  • Place a large shallow bowl on a Table and put a coin in it.
  • Move away slowly from the bowl. Stop when the coin just disappears from your sight.
  • Ask a friend to pour water gently into the bowl without disturbing the coin.
  • Q. Keep looking for the coin from your position. Does the coin becomes visible again from your position? How could this happen?
  • Answer : Yes, on pouring water into the bowl, the coin becomes visible again because due to refraction of light, for our eyes, the submerged coin apparently seems raised above its actual level and thus becomes visible on seeing from the same side and distance
  • Draw a thick straight line in ink, over a sheet of white paper placed on a Table.
  • Place a glass slab over the line in such a way that one of its edges makes an angle with the line.
  • Q. 1. Look at the portion of the line under the slab from the sides. What do you observe? Does the line under the glass slab appear to be bent at the edges?
  • Answer :Yes, due to the refration of light, the line under the glass slab appear to be bent at the edges
  • Q. 2. Next, place the glass slab such that it is normal to the line. What do you observe now? Does the part of the line under the glass slab appear bent?
  • Answer : No, Now the part of the line under the glass slab appear in a straight line. Because a ray of light, which is perpendicular to the plain of a refracting medium, does not change its angle due to refraction.
  • Q. 3. Look at the line from the top of the glass slab. Does the part of the line, beneath the slab, appear to be raised? Why does this happen?
  • Answer : Yes, the part of the line, beneath the slab, appear to be raised. Due to refraction of light, apparent position of image of object seems nearer than its actual position.
  • Fix a sheet of white paper on a drawing board using drawing pins.
  • Place a rectangular glass slab over the sheet in the middle.
  • Draw the outline of the slab with a pencil. Let us name the outline as ABCD.
  • Take four identical pins.
  • Fix two pins, say E and F, vertically such that the line joining the pins is inclined to the edge AB.
  • Look for the images of the pins E and F through the opposite edge. Fix two other pins, say G and H, such that these pins and the images of E and F lie on a straight line.
  • Remove the pins and the slab.
  • Join the positions of tip of the pins E and F and produce the line up to AB. Let EF meet AB at O. Similarly, join the positions of tip of the pins G and H and produce it up to the edge CD. Let HG meet CD at O′.
  • Join O and O′. Also produce EF up to P, as shown by a dotted line in Fig. 10.10.

Question 1. What happens to incident ray as it enters the glass slab ?

Answer : The incident ray as it enters from a rarer medium of air to a denser medium of glass, bends towards the normal, due to refraction of light . The refraction is cuased by change in the speed of light as it enters from one transparent medium to another.

Question 2. What happens to emergent ray as it leaves the glass slab ?

Answer : The emergent ray as it leaves the denser medium of glass and enters into a rarer medium of air, bends away from the normal, due to refraction of light. The refraction is caused by change in the speed of light as it enters from one transparent medium to another.

Question 3. What is the perpendicular distance between directions of incident and emergent rays ?

Answer : Lateral displacement. This gives a measure of path deviation of refracted rays due to refraction.

Question 4. As given in the activity above, the medium of incidence and emergent ray is same (air), what could be the possible observations for angle of incidence and angle of emergence ?

Answer : When the medium of incidence and emergent ray is same (air), angle of incidence is equal to angle of emergence.


Chapter 8: Content

"This postcard from the edge of reason came to feel like a developmental milestone, an instant of self-consciousness in which it became clear that I was undergoing a transformation. I was being freshly coded with certain expectations of the world, one of which seemed to be an unflagging belief in the responsiveness of others and which never seemed to learn from its disappointments. Digital technology was reshaping my responses, collaborating with my instincts, creating in me, its subject, all kinds of new sensitivities." – Laurence Scott

Chapter Summary

Content decisions boil down to three basic, interconnected questions: What are my content options? Who generates the content? How do I select the right content?

Content options have two distinct dimensions: form and type. Form options address, “Is the content primarily in the form of a picture, text, video or graphic?” Type of content options address what is the nature of what actually gets posted on social media sites. Categories include such items as: news & information, events, calls to action, amusements and commentaries.

You can create your own content (internally-produced), draw content from other sources that you re-purpose, (curated), create content with your followers’ input (co-created), or rely on your fans to develop content without any prompting (user-generated).

Effective social media managers select the right content by abiding by the following principles: a) the content is aligned with the coordinates, b) sensitive to the audiences, c) compatible with the channels, d) properly apportioned among categories, and e) routinely monitored.

Chapter Outline

  • What Are My Content Options?
  • Who Generates the Content?
    • Internally-produced content
    • Curated content
    • Co-created content
    • User-generated content
    • Coordinate aligned
    • Audience sensitive
    • Channel compatible
    • Category apportioned
    • Feedback driven

    Chapter Deep Dive Study Questions

    These exercises are designed to enhance your understanding of the chapter’s key ideas, principles and approaches.

    #1
    Rank order the principles in Figure 8.1 from least difficult to consistently follow (1) to most difficult (5). Provide your rationale.

    #2
    Find three recent examples of poor social media content decisions. Discuss which principles each decision violated.

    #3
    Construct a grid where the horizontal axis lists 5 social media platforms you are most familiar with. On the vertical axis, identify types of content categories. a) Using the grid, place a check by the social media platforms that seem most compatible with the content category. b) Place an X by those channels that are least compatible with the content category. c) Provide your rationale for your choices.


    PSY520 All Week Exercises November 2018

    Complete the following exercises from “Review Questions” located at the end of each chapter and put them into a Word document to be submitted as directed by the instructor.

    Chapter 1, numbers 1.8 and 1.9

    Chapter 2, numbers 2.14, 2.17, and 2.18

    Chapter 3, numbers 3.13, 3.14, 3.18, and 3.19

    Chapter 4, numbers 4.9, 4.14, 4.17, and 4.19

    Show all relevant work use the equation editor in Microsoft Word when necessary.

    PSY 520 Graduate Statistics

    Week 2 Exercises

    Complete the following exercises located at the end of each chapter and put them into a Word document to be submitted as directed by the instructor.

    Show all relevant work use the equation editor in Microsoft Word when necessary.

    Chapter 5, numbers 5.11, 5.13, 5.15, and 5.18

    Chapter 8, numbers 8.10, 8.14, 8.16, 8.19, and 8.21

    PSY 520 Graduate Statistics

    Week 3 Exercises

    Complete the following exercises located at the end of each chapter and put them into a Word document to be submitted as directed by the instructor.

    Show all relevant work use the equation editor in Microsoft Word when necessary.

    Chapter 6, numbers 6.7, 6.10, and 6.11

    Chapter 7, numbers 7.8, 7.10, and 7.13

    PSY 520 Graduate Statistics

    Week 4 Exercises

    Complete the following exercises located at the end of each chapter and put them into a Word document to be submitted as directed by the instructor.

    Show all relevant work use the equation editor in Microsoft Word when necessary.

    Chapter 9, numbers 9.7, 9.8, 9.9, 9.13, and 9.14

    Chapter 10, numbers 10.9, 10.10, 10.11, and 10.12

    Chapter 11, numbers 11.11, 11.19, and 11.20

    Chapter 12, numbers 12.7, 12.8, and 12.10

    PSY 520 Graduate Statistics

    Week 5 Exercises

    Complete the following exercises located at the end of each chapter and put them into a Word document to be submitted as directed by the instructor.

    Show all relevant work use the equation editor in Microsoft Word when necessary.

    Chapter 13, numbers 13.6, 13.8, 13.9, and 13.10

    Chapter 14, numbers 14.11, 14.12, and 14.14

    Chapter 15, numbers 15.7, 15.8, 15.10 and 15.14

    PSY 520 Graduate Statistics

    Week 6 Exercises

    Complete the following exercises located at the end of each chapter and put them into a Word document to be submitted as directed by the instructor.

    Show all relevant work use the equation editor in Microsoft Word when necessary.

    Chapter 16, numbers 16.9, 16.10, 16.12 and 16.14

    Chapter 17, numbers 17.6, 17.7, and 17.8

    Chapter 18, numbers 18.8, 18.11, and 18.12

    PSY 520 Graduate Statistics

    Week 7 Exercises

    Complete the following exercises located at the end of each chapter and put them into a Word document to be submitted as directed by the instructor.

    Show all relevant work use the equation editor in Microsoft Word when necessary.

    Chapter 19, numbers 19.9, 19.10, 19.13, 19.14, and 19.16

    Chapter 20, numbers 20.5, 20.6, 20.7, and 20.10

    PSY 520 Graduate Statistics

    Week 8 Exercises

    Complete the following exercises located at the end of each chapter and put them into a Word document to be submitted as directed by the instructor.

    Show all relevant work use the equation editor in Microsoft Word when necessary.


    8 Chapter Review

    The kinetic energy of a system must always be positive or zero. Explain whether this is true for the potential energy of a system.

    The force exerted by a diving board is conservative, provided the internal friction is negligible. Assuming friction is negligible, describe changes in the potential energy of a diving board as a swimmer drives from it, starting just before the swimmer steps on the board until just after his feet leave it.

    Describe the gravitational potential energy transfers and transformations for a javelin, starting from the point at which an athlete picks up the javelin and ending when the javelin is stuck into the ground after being thrown.

    A couple of soccer balls of equal mass are kicked off the ground at the same speed but at different angles. Soccer ball A is kicked off at an angle slightly above the horizontal, whereas ball B is kicked slightly below the vertical. How do each of the following compare for ball A and ball B? (a) The initial kinetic energy and (b) the change in gravitational potential energy from the ground to the highest point? If the energy in part (a) differs from part (b), explain why there is a difference between the two energies.

    What is the dominant factor that affects the speed of an object that started from rest down a frictionless incline if the only work done on the object is from gravitational forces?

    Two people observe a leaf falling from a tree. One person is standing on a ladder and the other is on the ground. If each person were to compare the energy of the leaf observed, would each person find the following to be the same or different for the leaf, from the point where it falls off the tree to when it hits the ground: (a) the kinetic energy of the leaf (b) the change in gravitational potential energy (c) the final gravitational potential energy?

    8.2 Conservative and Non-Conservative Forces

    What is the physical meaning of a non-conservative force?

    A bottle rocket is shot straight up in the air with a speed 30 m/s 30m/s . If the air resistance is ignored, the bottle would go up to a height of approximately 46 m 46m . However, the rocket goes up to only 35 m 35m before returning to the ground. What happened? Explain, giving only a qualitative response.

    An external force acts on a particle during a trip from one point to another and back to that same point. This particle is only effected by conservative forces. Does this particle’s kinetic energy and potential energy change as a result of this trip?

    8.3 Conservation of Energy

    When a body slides down an inclined plane, does the work of friction depend on the body’s initial speed? Answer the same question for a body sliding down a curved surface.

    Consider the following scenario. A car for which friction is not negligible accelerates from rest down a hill, running out of gasoline after a short distance (see below). The driver lets the car coast farther down the hill, then up and over a small crest. He then coasts down that hill into a gas station, where he brakes to a stop and fills the tank with gasoline. Identify the forms of energy the car has, and how they are changed and transferred in this series of events.

    A dropped ball bounces to one-half its original height. Discuss the energy transformations that take place.

    “ E = K + U E=K+U constant is a special case of the work-energy theorem.” Discuss this statement.

    In a common physics demonstration, a bowling ball is suspended from the ceiling by a rope.

    The professor pulls the ball away from its equilibrium position and holds it adjacent to his nose, as shown below. He releases the ball so that it swings directly away from him. Does he get struck by the ball on its return swing? What is he trying to show in this demonstration?

    A child jumps up and down on a bed, reaching a higher height after each bounce. Explain how the child can increase his maximum gravitational potential energy with each bounce.

    Can a non-conservative force increase the mechanical energy of the system?

    Neglecting air resistance, how much would I have to raise the vertical height if I wanted to double the impact speed of a falling object?

    A box is dropped onto a spring at its equilibrium position. The spring compresses with the box attached and comes to rest. Since the spring is in the vertical position, does the change in the gravitational potential energy of the box while the spring is compressing need to be considered in this problem?

    Problems

    8.1 Potential Energy of a System

    Using values from Table 8.2, how many DNA molecules could be broken by the energy carried by a single electron in the beam of an old-fashioned TV tube? (These electrons were not dangerous in themselves, but they did create dangerous X-rays. Later-model tube TVs had shielding that absorbed X-rays before they escaped and exposed viewers.)

    If the energy in fusion bombs were used to supply the energy needs of the world, how many of the 9-megaton variety would be needed for a year’s supply of energy (using data from Table 8.1)?

    A camera weighing 10 N falls from a small drone hovering 20 m 20m overhead and enters free fall. What is the gravitational potential energy change of the camera from the drone to the ground if you take a reference point of (a) the ground being zero gravitational potential energy? (b) The drone being zero gravitational potential energy? What is the gravitational potential energy of the camera (c) before it falls from the drone and (d) after the camera lands on the ground if the reference point of zero gravitational potential energy is taken to be a second person looking out of a building 30 m 30m from the ground?

    Someone drops a 50 − g 50−g pebble off of a docked cruise ship, 70.0 m 70.0m from the water line. A person on a dock 3.0 m 3.0m from the water line holds out a net to catch the pebble. (a) How much work is done on the pebble by gravity during the drop? (b) What is the change in the gravitational potential energy during the drop? If the gravitational potential energy is zero at the water line, what is the gravitational potential energy (c) when the pebble is dropped? (d) When it reaches the net? What if the gravitational potential energy was 30.0 30.0 Joules at water level? (e) Find the answers to the same questions in (c) and (d).

    A cat’s crinkle ball toy of mass 15 g 15g is thrown straight up with an initial speed of 3 m/s 3m/s . Assume in this problem that air drag is negligible. (a) What is the kinetic energy of the ball as it leaves the hand? (b) How much work is done by the gravitational force during the ball’s rise to its peak? (c) What is the change in the gravitational potential energy of the ball during the rise to its peak? (d) If the gravitational potential energy is taken to be zero at the point where it leaves your hand, what is the gravitational potential energy when it reaches the maximum height? (e) What if the gravitational potential energy is taken to be zero at the maximum height the ball reaches, what would the gravitational potential energy be when it leaves the hand? (f) What is the maximum height the ball reaches?

    8.2 Conservative and Non-Conservative Forces

    A force F ( x ) = ( 3.0 / x ) N F(x)=(3.0/x)N acts on a particle as it moves along the positive x-axis. (a) How much work does the force do on the particle as it moves from x = 2.0 m x=2.0m to x = 5.0 m? x=5.0m? (b) Picking a convenient reference point of the potential energy to be zero at x = ∞ , x=∞, find the potential energy for this force.

    A force F ( x ) = ( −5.0 x 2 + 7.0 x ) N F(x)=(−5.0ࡨ+7.0x)N acts on a particle. (a) How much work does the force do on the particle as it moves from x = 2.0 m x=2.0m to x = 5.0 m? x=5.0m? (b) Picking a convenient reference point of the potential energy to be zero at x = ∞ , x=∞, find the potential energy for this force.

    Find the force corresponding to the potential energy U ( x ) = − a / x + b / x 2 . U(x)=−a/x+b/x2.

    The potential energy function for either one of the two atoms in a diatomic molecule is often approximated by U ( x ) = − a / x 12 − b / x 6 U(x)=−a/x12−b/x6 where x is the distance between the atoms. (a) At what distance of seperation does the potential energy have a local minimum (not at x = ∞ ) ? x=∞)? (b) What is the force on an atom at this separation? (c) How does the force vary with the separation distance?

    A particle of mass 2.0 kg 2.0kg moves under the influence of the force F ( x ) = ( 3 / x √ ) N . F(x)=(3/x)N. If its speed at x = 2.0 m x=2.0m is v = 6.0 m/s, v=6.0m/s, what is its speed at x = 7.0 m? x=7.0m?

    A particle of mass 2.0 kg 2.0kg moves under the influence of the force F ( x ) = ( −5 x 2 + 7 x ) N . F(x)=(−5ࡨ+7x)N. If its speed at x = −4.0 m x=−4.0m is v = 20.0 m/s, v=20.0m/s, what is its speed at x = 4.0 m ? x=4.0m?

    A crate on rollers is being pushed without frictional loss of energy across the floor of a freight car (see the following figure). The car is moving to the right with a constant speed v 0 . v0. If the crate starts at rest relative to the freight car, then from the work-energy theorem, F d = m v 2 / 2 , Fd=mv2/2, where d, the distance the crate moves, and v, the speed of the crate, are both measured relative to the freight car. (a) To an observer at rest beside the tracks, what distance d ′ d′ is the crate pushed when it moves the distance d in the car? (b) What are the crate’s initial and final speeds v 0 ′ v0′ and v ′ v′ as measured by the observer beside the tracks? (c) Show that F d ′ = m ( v ′ ) 2 / 2 − m ( v ′ 0 ) 2 / 2 Fd′=m(v′)2/2−m(v′0)2/2 and, consequently, that work is equal to the change in kinetic energy in both reference systems.

    8.3 Conservation of Energy

    A boy throws a ball of mass 0.25 kg 0.25kg straight upward with an initial speed of 20 m / s 20m/s When the ball returns to the boy, its speed is 17 m / s 17m/s How much much work does air resistance do on the ball during its flight?

    A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance?

    Using energy considerations and assuming negligible air resistance, show that a rock thrown from a bridge 20.0 m above water with an initial speed of 15.0 m/s strikes the water with a speed of 24.8 m/s independent of the direction thrown. (Hint:show that K i + U i = K f + U f ) Ki+Ui=Kf+Uf)

    A 1.0-kg ball at the end of a 2.0-m string swings in a vertical plane. At its lowest point the ball is moving with a speed of 10 m/s. (a) What is its speed at the top of its path? (b) What is the tension in the string when the ball is at the bottom and at the top of its path?

    Ignoring details associated with friction, extra forces exerted by arm and leg muscles, and other factors, we can consider a pole vault as the conversion of an athlete’s running kinetic energy to gravitational potential energy. If an athlete is to lift his body 4.8 m during a vault, what speed must he have when he plants his pole?

    Tarzan grabs a vine hanging vertically from a tall tree when he is running at 9.0 m / s . 9.0m/s. (a) How high can he swing upward? (b) Does the length of the vine affect this height?

    Assume that the force of a bow on an arrow behaves like the spring force. In aiming the arrow, an archer pulls the bow back 50 cm and holds it in position with a force of 150 N 150N . If the mass of the arrow is 50 g 50g and the “spring” is massless, what is the speed of the arrow immediately after it leaves the bow?

    A 100 − kg 100−kg man is skiing across level ground at a speed of 8.0 m/s 8.0m/s when he comes to the small slope 1.8 m higher than ground level shown in the following figure. (a) If the skier coasts up the hill, what is his speed when he reaches the top plateau? Assume friction between the snow and skis is negligible. (b) What is his speed when he reaches the upper level if an 80 − N 80−N frictional force acts on the skis?

    A sled of mass 70 kg starts from rest and slides down a 10 ° 10° incline 80 m 80m long. It then travels for 20 m horizontally before starting back up an 8 ° 8° incline. It travels 80 m along this incline before coming to rest. What is the net work done on the sled by friction?

    A girl on a skateboard (total mass of 40 kg) is moving at a speed of 10 m/s at the bottom of a long ramp. The ramp is inclined at 20 ° 20° with respect to the horizontal. If she travels 14.2 mupward along the ramp before stopping, what is the net frictional force on her?

    A baseball of mass 0.25 kg is hit at home plate with a speed of 40 m/s. When it lands in a seat in the left-field bleachers a horizontal distance 120 m from home plate, it is moving at 30 m/s. If the ball lands 20 m above the spot where it was hit, how much work is done on it by air resistance?

    A small block of mass m slides without friction around the loop-the-loop apparatus shown below. (a) If the block starts from rest at A, what is its speed at B? (b) What is the force of the track on the block at B?

    The massless spring of a spring gun has a force constant k = 12 N/cm . k=12N/cm. When the gun is aimed vertically, a 15-g projectile is shot to a height of 5.0 m above the end of the expanded spring. (See below.) How much was the spring compressed initially?

    A small ball is tied to a string and set rotating with negligible friction in a vertical circle. Prove that the tension in the string at the bottom of the circle exceeds that at the top of the circle by eight times the weight of the ball. Assume the ball’s speed is zero as it sails over the top of the circle and there is no additional energy added to the ball during rotation.

    8.4 Potential Energy Diagrams and Stability

    A mysterious constant force of 10 N acts horizontally on everything. The direction of the force is found to be always pointed toward a wall in a big hall. Find the potential energy of a particle due to this force when it is at a distance x from the wall, assuming the potential energy at the wall to be zero.

    A single force F ( x ) = −4.0 x F(x)=−4.0x (in newtons) acts on a 1.0-kg body. When x = 3.5 m, x=3.5m, the speed of the body is 4.0 m/s. What is its speed at x = 2.0 m? x=2.0m?

    A particle of mass 4.0 kg is constrained to move along the x-axis under a single force F ( x ) = − c x 3 , F(x)=−cx3, where c = 8.0 N/m 3 . c=8.0N/m3. The particle’s speed at A, where x A = 1.0 m, xA=1.0m, is 6.0 m/s. What is its speed at B, where x B = −2.0 m? xB=−2.0m?

    The force on a particle of mass 2.0 kg varies with position according to F ( x ) = −3.0 x 2 F(x)=−3.0ࡨ (x in meters, F(x) in newtons). The particle’s velocity at x = 2.0 m x=2.0m is 5.0 m/s. Calculate the mechanical energy of the particle using (a) the origin as the reference point and (b) x = 4.0 m x=4.0m as the reference point. (c) Find the particle’s velocity at x = 1.0 m . x=1.0m. Do this part of the problem for each reference point.

    A 4.0-kg particle moving along the x-axis is acted upon by the force whose functional form appears below. The velocity of the particle at x = 0 x=0 is v = 6.0 m/s . v=6.0m/s. Find the particle’s speed at x = ( a ) 2.0 m , ( b ) 4.0 m , ( c ) 10.0 m , ( d ) x=(a)2.0m,(b)4.0m,(c)10.0m,(d) Does the particle turn around at some point and head back toward the origin? (e) Repeat part (d) if v = 2.0 m/s at x = 0 . v=2.0m/satx=0.

    A particle of mass 0.50 kg moves along the x-axis with a potential energy whose dependence on x is shown below. (a) What is the force on the particle at x = 2.0 , 5.0 , 8.0 , and x=2.0,5.0,8.0,and 12 m? (b) If the total mechanical energy E of the particle is −6.0 J, what are the minimum and maximum positions of the particle? (c) What are these positions if E = 2.0 J? E=2.0J? (d) If E = 16 J E=16J , what are the speeds of the particle at the positions listed in part (a)?

    (a) Sketch a graph of the potential energy function U ( x ) = k x 2 / 2 + A e − α x 2 , U(x)=kx2/2+Ae−αx2, where k , A , and α k,A,andα are constants. (b) What is the force corresponding to this potential energy? (c) Suppose a particle of mass m moving with this potential energy has a velocity v a va when its position is x = a x=a . Show that the particle does not pass through the origin unless

    8.5 Sources of Energy

    In the cartoon movie Pocahontas, Pocahontas runs to the edge of a cliff and jumps off, showcasing the fun side of her personality. (a) If she is running at 3.0 m/s before jumping off the cliff and she hits the water at the bottom of the cliff at 20.0 m/s, how high is the cliff? Assume negligible air drag in this cartoon. (b) If she jumped off the same cliff from a standstill, how fast would she be falling right before she hit the water?

    In the reality television show “Amazing Race”, a contestant is firing 12-kg watermelons from a slingshot to hit targets down the field. The slingshot is pulled back 1.5 m and the watermelon is considered to be at ground level. The launch point is 0.3 m from the ground and the targets are 10 m horizontally away. Calculate the spring constant of the slingshot.

    In the Back to the Future movies, a DeLorean car of mass 1230 kg travels at 88 miles per hour to venture back to the future. (a) What is the kinetic energy of the DeLorian? (b) What spring constant would be needed to stop this DeLorean in a distance of 0.1m?

    In the Hunger Games movie, Katniss Everdeen fires a 0.0200-kg arrow from ground level to pierce an apple up on a stage. The spring constant of the bow is 330 N/m and she pulls the arrow back a distance of 0.55 m. The apple on the stage is 5.00 m higher than the launching point of the arrow. At what speed does the arrow (a) leave the bow? (b) strike the apple?

    In a “Top Fail” video, two women run at each other and collide by hitting exercise balls together. If each woman has a mass of 50 kg, which includes the exercise ball, and one woman runs to the right at 2.0 m/s and the other is running toward her at 1.0 m/s, (a) how much total kinetic energy is there in the system? (b) If energy is conserved after the collision and each exercise ball has a mass of 2.0 kg, how fast would the balls fly off toward the camera?

    In a Coyote/Road Runner cartoon clip, a spring expands quickly and sends the coyote into a rock. If the spring extended 5 m and sent the coyote of mass 20 kg to a speed of 15 m/s, (a) what is the spring constant of this spring? (b) If the coyote were sent vertically into the air with the energy given to him by the spring, how high could he go if there were no non-conservative forces?

    In an iconic movie scene, Forrest Gump runs around the country. If he is running at a constant speed of 3 m/s, would it take him more or less energy to run uphill or downhill and why?

    In the movie Monty Python and the Holy Grail a cow is catapulted from the top of a castle wall over to the people down below. The gravitational potential energy is set to zero at ground level. The cow is launched from a spring of spring constant 1.1 × 10 4 N/m 1.1×104N/m that is expanded 0.5 m from equilibrium. If the castle is 9.1 m tall and the mass of the cow is 110 kg, (a) what is the gravitational potential energy of the cow at the top of the castle? (b) What is the elastic spring energy of the cow before the catapult is released? (c) What is the speed of the cow right before it lands on the ground?

    A 60.0-kg skier with an initial speed of 12.0 m/s coasts up a 2.50-m high rise as shown. Find her final speed at the top, given that the coefficient of friction between her skis and the snow is 0.80.

    (a) How high a hill can a car coast up (engines disengaged) if work done by friction is negligible and its initial speed is 110 km/h? (b) If, in actuality, a 750-kg car with an initial speed of 110 km/h is observed to coast up a hill to a height 22.0 m above its starting point, how much thermal energy was generated by friction? (c) What is the average force of friction if the hill has a slope of 2.5 ° 2.5° above the horizontal?

    A 5.00 × 10 5 -kg 5.00×105-kg subway train is brought to a stop from a speed of 0.500 m/s in 0.400 m by a large spring bumper at the end of its track. What is the spring constant k of the spring?

    A pogo stick has a spring with a spring constant of 2.5 × 10 4 N/m, 2.5×104N/m, which can be compressed 12.0 cm. To what maximum height from the uncompressed spring can a child jump on the stick using only the energy in the spring, if the child and stick have a total mass of 40 kg?

    A block of mass 500 g is attached to a spring of spring constant 80 N/m (see the following figure). The other end of the spring is attached to a support while the mass rests on a rough surface with a coefficient of friction of 0.20 that is inclined at angle of 30 ° . 30°. The block is pushed along the surface till the spring compresses by 10 cm and is then released from rest. (a) How much potential energy was stored in the block-spring-support system when the block was just released? (b) Determine the speed of the block when it crosses the point when the spring is neither compressed nor stretched. (c) Determine the position of the block where it just comes to rest on its way up the incline.

    A block of mass 200 g is attached at the end of a massless spring of spring constant 50 N/m. The other end of the spring is attached to the ceiling and the mass is released at a height considered to be where the gravitational potential energy is zero. (a) What is the net potential energy of the block at the instant the block is at the lowest point? (b) What is the net potential energy of the block at the midpoint of its descent? (c) What is the speed of the block at the midpoint of its descent?

    A T-shirt cannon launches a shirt at 5.00 m/s from a platform height of 3.00 m from ground level. How fast will the shirt be traveling if it is caught by someone whose hands are (a) 1.00 m from ground level? (b) 4.00 m from ground level? Neglect air drag.

    A child (32 kg) jumps up and down on a trampoline. The trampoline exerts a spring restoring force on the child with a constant of 5000 N/m. At the highest point of the bounce, the child is 1.0 m above the level surface of the trampoline. What is the compression distance of the trampoline? Neglect the bending of the legs or any transfer of energy of the child into the trampoline while jumping.

    Shown below is a box of mass m 1 m1 that sits on a frictionless incline at an angle above the horizontal θ θ . This box is connected by a relatively massless string, over a frictionless pulley, and finally connected to a box at rest over the ledge, labeled m 2 m2 . If m 1 m1 and m 2 m2 are a height h above the ground and m 2 >> m 1 m2>>m1 : (a) What is the initial gravitational potential energy of the system? (b) What is the final kinetic energy of the system?

    Additional Problems

    A massless spring with force constant k = 200 N/m k=200N/m hangs from the ceiling. A 2.0-kg block is attached to the free end of the spring and released. If the block falls 17 cm before starting back upwards, how much work is done by friction during its descent?

    A particle of mass 2.0 kg moves under the influence of the force F ( x ) = ( −5 x 2 + 7 x ) N . F(x)=(−5ࡨ+7x)N. Suppose a frictional force also acts on the particle. If the particle’s speed when it starts at x = −4.0 m x=−4.0m is 0.0 m/s and when it arrives at x = 4.0 m x=4.0m is 9.0 m/s, how much work is done on it by the frictional force between x = −4.0 m x=−4.0m and x = 4.0 m? x=4.0m?

    Block 2 shown below slides along a frictionless table as block 1 falls. Both blocks are attached by a frictionless pulley. Find the speed of the blocks after they have each moved 2.0 m. Assume that they start at rest and that the pulley has negligible mass. Use m 1 = 2.0 kg m1=2.0kg and m 2 = 4.0 kg . m2=4.0kg.

    A body of mass m and negligible size starts from rest and slides down the surface of a frictionless solid sphere of radius R. (See below.) Prove that the body leaves the sphere when θ = cos −1 ( 2 / 3 ) . θ=cos−1(2/3).

    A mysterious force acts on all particles along a particular line and always points towards a particular point P on the line. The magnitude of the force on a particle increases as the cube of the distance from that point that is F ∞ r 3 F∞r3 , if the distance from P to the position of the particle is r. Let b be the proportionality constant, and write the magnitude of the force as F = b r 3 F=br3 . Find the potential energy of a particle subjected to this force when the particle is at a distance D from P, assuming the potential energy to be zero when the particle is at P.

    An object of mass 10 kg is released at point A, slides to the bottom of the 30 ° 30° incline, then collides with a horizontal massless spring, compressing it a maximum distance of 0.75 m. (See below.) The spring constant is 500 M/m, the height of the incline is 2.0 m, and the horizontal surface is frictionless. (a) What is the speed of the object at the bottom of the incline? (b) What is the work of friction on the object while it is on the incline? (c) The spring recoils and sends the object back toward the incline. What is the speed of the object when it reaches the base of the incline? (d) What vertical distance does it move back up the incline?

    Shown below is a small ball of mass m attached to a string of length a. A small peg is located a distance h below the point where the string is supported. If the ball is released when the string is horizontal, show that h must be greater than 3a/5 if the ball is to swing completely around the peg.

    A block leaves a frictionless inclined surfarce horizontally after dropping off by a height h. Find the horizontal distance Dwhere it will land on the floor, in terms of h, H, and g.

    A block of mass m, after sliding down a frictionless incline, strikes another block of mass M that is attached to a spring of spring constant k (see below). The blocks stick together upon impact and travel together. (a) Find the compression of the spring in terms of m, M, h, g, and k when the combination comes to rest. Hint: The speed of the combined blocks m + M ( v 2 ) m+M(v2) is based on the speed of block m just prior to the collision with the block M (v1) based on the equation v 2 = ( m / m ) + M ( v 1 ) v2=(m/m)+M(v1) . This will be discussed further in the chapter on Linear Momentum and Collisions. (b) The loss of kinetic energy as a result of the bonding of the two masses upon impact is stored in the so-called binding energy of the two masses. Calculate the binding energy.

    A block of mass 300 g is attached to a spring of spring constant 100 N/m. The other end of the spring is attached to a support while the block rests on a smooth horizontal table and can slide freely without any friction. The block is pushed horizontally till the spring compresses by 12 cm, and then the block is released from rest. (a) How much potential energy was stored in the block-spring support system when the block was just released? (b) Determine the speed of the block when it crosses the point when the spring is neither compressed nor stretched. (c) Determine the speed of the block when it has traveled a distance of 20 cm from where it was released.

    Consider a block of mass 0.200 kg attached to a spring of spring constant 100 N/m. The block is placed on a frictionless table, and the other end of the spring is attached to the wall so that the spring is level with the table. The block is then pushed in so that the spring is compressed by 10.0 cm. Find the speed of the block as it crosses (a) the point when the spring is not stretched, (b) 5.00 cm to the left of point in (a), and (c) 5.00 cm to the right of point in (a).

    A skier starts from rest and slides downhill. What will be the speed of the skier if he drops by 20 meters in vertical height? Ignore any air resistance (which will, in reality, be quite a lot), and any friction between the skis and the snow.

    Repeat the preceding problem, but this time, suppose that the work done by air resistance cannot be ignored. Let the work done by the air resistance when the skier goes from A to B along the given hilly path be −2000 J. The work done by air resistance is negative since the air resistance acts in the opposite direction to the displacement. Supposing the mass of the skier is 50 kg, what is the speed of the skier at point B?

    Two bodies are interacting by a conservative force. Show that the mechanical energy of an isolated system consisting of two bodies interacting with a conservative force is conserved. (Hint: Start by using Newton’s third law and the definition of work to find the work done on each body by the conservative force.)

    In an amusement park, a car rolls in a track as shown below. Find the speed of the car at A, B, and C. Note that the work done by the rolling friction is zero since the displacement of the point at which the rolling friction acts on the tires is momentarily at rest and therefore has a zero displacement.

    A 200-g steel ball is tied to a 2.00-m “massless” string and hung from the ceiling to make a pendulum, and then, the ball is brought to a position making a 30 ° 30° angle with the vertical direction and released from rest. Ignoring the effects of the air resistance, find the speed of the ball when the string (a) is vertically down, (b) makes an angle of 20 ° 20° with the vertical and (c) makes an angle of 10 ° 10° with the vertical.

    A hockey puck is shot across an ice-covered pond. Before the hockey puck was hit, the puck was at rest. After the hit, the puck has a speed of 40 m/s. The puck comes to rest after going a distance of 30 m. (a) Describe how the energy of the puck changes over time, giving the numerical values of any work or energy involved. (b) Find the magnitude of the net friction force.

    A projectile of mass 2 kg is fired with a speed of 20 m/s at an angle of 30 ° 30° with respect to the horizontal. (a) Calculate the initial total energy of the projectile given that the reference point of zero gravitational potential energy at the launch position. (b) Calculate the kinetic energy at the highest vertical position of the projectile. (c) Calculate the gravitational potential energy at the highest vertical position. (d) Calculate the maximum height that the projectile reaches. Compare this result by solving the same problem using your knowledge of projectile motion.

    An artillery shell is fired at a target 200 m above the ground. When the shell is 100 m in the air, it has a speed of 100 m/s. What is its speed when it hits its target? Neglect air friction.

    How much energy is lost to a dissipative drag force if a 60-kg person falls at a constant speed for 15 meters?

    A box slides on a frictionless surface with a total energy of 50 J. It hits a spring and compresses the spring a distance of 25 cm from equilibrium. If the same box with the same initial energy slides on a rough surface, it only compresses the spring a distance of 15 cm, how much energy must have been lost by sliding on the rough surface?


    Calculus: Graphical, Numerical, Algebraic, 3rd Edition Answers Ch 1 Prerequisites for Calculus Ex 1.1

    Chapter 1 Prerequisites for Calculus Exercise 1.1 1E
    The given coordinates are A(1, 2) and B(-1, -1)
    Now to find the increments in coordinate we subtract one coordinate point from the other, as shown below:
    The increment in x-coordinate is:

    Chapter 1 Prerequisites for Calculus Exercise 1.1 1QR

    Chapter 1 Prerequisites for Calculus Exercise 1.1 2E
    The given coordinates are A(3, 2) and B(-1, -2)
    Now to find the increments in coordinate we subtract one coordinate point from the other, as shown below:
    The increment in x-coordinate is:

    Chapter 1 Prerequisites for Calculus Exercise 1.1 2QR

    Chapter 1 Prerequisites for Calculus Exercise 1.1 3E
    The given coordinates are A(-3, 1) and B(-8, -1)
    Now to find the increments in coordinate we subtract one coordinate point from the other, as shown below:
    The increment in x-coordinate is:

    Chapter 1 Prerequisites for Calculus Exercise 1.1 3QR

    Chapter 1 Prerequisites for Calculus Exercise 1.1 4E
    The given coordinates are A(0, 4) and B(0, -2)
    Now to find the increments in coordinate we subtract one coordinate point from the other, as shown below:
    The increment in x-coordinate is:

    Chapter 1 Prerequisites for Calculus Exercise 1.1 4QR

    Chapter 1 Prerequisites for Calculus Exercise 1.1 5E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 5QR
    The given equation is 3(x) – 4(y) = 5
    (a) Replacing the given ordered pair in the equation, as below,

    Chapter 1 Prerequisites for Calculus Exercise 1.1 6E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 6QR
    The given equation is y = -2x + 5

    Chapter 1 Prerequisites for Calculus Exercise 1.1 7E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 7QR
    The given points are (1, 0) and (0, 1)
    The distance between two points is given as

    Chapter 1 Prerequisites for Calculus Exercise 1.1 8E


    Chapter 1 Prerequisites for Calculus Exercise 1.1 8QR

    Chapter 1 Prerequisites for Calculus Exercise 1.1 9E
    The given point is P(3, 2)
    (a) As we know that tor a vertical line we nave m = ∞ (undefined)
    In case ot a vertical line we can see that the x-coordinate does not change but remains constant.
    Therefore, the required equation is, x = 3

    Chapter 1 Prerequisites for Calculus Exercise 1.1 9QR
    The given equation is
    4x – 3y = 7
    Now as required in the question, the value ot y in terms ot x can be calculated as below:

    Chapter 1 Prerequisites for Calculus Exercise 1.1 10E



    Chapter 1 Prerequisites for Calculus Exercise 1.1 10QR
    The given equation is
    —2x + 5y = —3
    Now as required in the question, the value of y in terms of x can be calculated as below:

    Chapter 1 Prerequisites for Calculus Exercise 1.1 11E
    The given point is P(0, -√2)
    (a) As we know that tor a vertical line we have m = ∞(undefined)
    In case ot a vertical line we can see that the x-coordinate does not change but remains constant,
    Therefore, the required equation is, x = 0
    This equation is also the equation tor the y-axis

    Chapter 1 Prerequisites for Calculus Exercise 1.1 12E
    The given point is P(-π, 0)
    (a) As we know that tor a vertical line we nave m = ∞(undefined)
    In case ot a vertical line we can see that the x-coordinate does not change but remains constant,
    Therefore, the required equation is x = -π,

    Chapter 1 Prerequisites for Calculus Exercise 1.1 13E
    The given point is P(1, 1) and slope is m = 1

    Chapter 1 Prerequisites for Calculus Exercise 1.1 14E
    The given point is P(-1, 1) and slope is m = -1

    Chapter 1 Prerequisites for Calculus Exercise 1.1 15E
    The given point is P(0, 3) and slope is m = 2

    Chapter 1 Prerequisites for Calculus Exercise 1.1 16E
    The given point is P(-4, 0) and slope is m = -2

    Chapter 1 Prerequisites for Calculus Exercise 1.1 17E
    The given point is slope is m = 3 and the intercept b = -2
    As we know that the slope-intercept torm ot equation is given by
    y = mx + b
    Where,
    m is the slope ot the line,
    b is the given intercept ot the line,

    Chapter 1 Prerequisites for Calculus Exercise 1.1 18E
    The given point is slope is m = -1 and the intercept b = 2
    As we know that the slope-intercept torm ot equation is given by
    y = mx + b
    Where,
    m is the slope ot the line,
    b is the given intercept ot the line,

    Chapter 1 Prerequisites for Calculus Exercise 1.1 19E
    The given point is slope is m = -1/2 and the intercept b = -3
    As we know that the slope-intercept torm ot equation is given by
    y = mx + b
    Where,
    m is the slope ot the line,
    b is the given intercept ot the line,

    Chapter 1 Prerequisites for Calculus Exercise 1.1 20E
    The given point is slope is m = 1/3 and the intercept b = -1
    As we know that the slope-intercept torm ot equation is given by
    y = mx + b
    Where,
    m is the slope ot the line,
    b is the given intercept ot the line,

    Chapter 1 Prerequisites for Calculus Exercise 1.1 21E
    The given points on the required line are (0, 0) and (2, 3)
    We know that the general torm ot the line is given as,
    Ax+By=C
    Where A and B are nonzero terms.

    Chapter 1 Prerequisites for Calculus Exercise 1.1 22E
    The given points on the required line are (1, 1) and (2, 1)
    We know that the general torm ot the line is given as,
    Ax+By=C
    Where A and B are nonzero terms.

    Chapter 1 Prerequisites for Calculus Exercise 1.1 23E
    The given points on the required line are (-2, 0) and (-2, -2)
    We know that the general torm ot the line is given as,
    Ax+By=C
    Where A and B are nonzero terms.

    Chapter 1 Prerequisites for Calculus Exercise 1.1 24E
    The given points on the required line are (-2, 0) and (-2, -2)
    We know that the general torm ot the line is given as,
    Ax+By=C
    Where A and B are nonzero terms.

    Chapter 1 Prerequisites for Calculus Exercise 1.1 25E
    As given in the question, trom the grapn we see that the line contains the two points are (0, 0) and (10, 25)

    Chapter 1 Prerequisites for Calculus Exercise 1.1 26E
    As given in the question, trom the grapn we see that the line contains the two points are (0, 0) and (5, 2)

    Chapter 1 Prerequisites for Calculus Exercise 1.1 27E

    (b) On comparing the two equations we get
    The y-intercept of the line is b = 3
    (c) The graphical representation ot the line will be as shown below
    [-10,10] by [-10,10]

    Chapter 1 Prerequisites for Calculus Exercise 1.1 28E
    The given equation of the line is,
    x+y=2
    This equation can be restructured as,
    y=-x+2
    Therefore this equation can be compared witn the slope-intercept for of the equation given as
    y = mx + b
    (a) On comparing the two equations we get
    The slope of the line is m = -1
    (b) On comparing the two equations we get
    The y-intercept of the line is b = 2
    (c) The graphical representation ot the line will be as shown below
    [-10,10] by [-10,10]

    Chapter 1 Prerequisites for Calculus Exercise 1.1 29E

    (b) On comparing the two equations we get
    The y-intercept of the line is b = 4
    (c) The graphical representation ot the line will be as shown below
    [-10,10] by [-10,10]

    Chapter 1 Prerequisites for Calculus Exercise 1.1 30E
    The given equation of the line is,
    y=2x+4
    Therefore this equation can be compared witn the slope-intercept for of the equation given as
    y = mx + b
    (a) On comparing the two equations we get
    The slope of the line is m = 2
    (b) On comparing the two equations we get
    The y-intercept of the line is b = 2
    (c) The graphical representation ot the line will be as shown below
    [-10,10] by [-10,10]

    Chapter 1 Prerequisites for Calculus Exercise 1.1 31E
    The given equation of the line is,
    y=-x+2
    And the given point is P(0, 0)
    Therefore this equation can be compared witn the slope-intercept for of the equation given as
    y = mx + b
    On comparing the two equations we get
    The slope ottne line is m = -1

    Chapter 1 Prerequisites for Calculus Exercise 1.1 32E
    The given equation of the line is,
    2x+x=4
    And the given point is P(-2, 2)
    Therefore this equation can be compared witn the slope-intercept for of the equation given as
    y = mx + b
    On comparing the two equations we get
    The slope ottne line is m = -2

    Chapter 1 Prerequisites for Calculus Exercise 1.1 33E
    The given equation of the line is,
    x=5
    And the given point is P(-2, 4)
    Therefore this equation can be compared witn the slope-intercept for of the equation given as
    y = mx + b
    On comparing the two equations we get
    The slope ottne line is undefined as the line is vertical.

    Chapter 1 Prerequisites for Calculus Exercise 1.1 34E
    The given equation of the line is,
    y=3
    And the given point is P(-1, 1/2)
    Therefore this equation can be compared witn the slope-intercept for of the equation given as
    y = mx + b
    On comparing the two equations we get
    The slope ottne line is m=0

    Chapter 1 Prerequisites for Calculus Exercise 1.1 35E
    The given function is
    f(x) = mx + b



    Chapter 1 Prerequisites for Calculus Exercise 1.1 36E
    The given function is
    f(x) = mx + b



    Chapter 1 Prerequisites for Calculus Exercise 1.1 37E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 38E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 39E
    As asked in the question, we have to write the slope intercept form of the equation for the line passing through the points (—2, —1) and (3, 4)

    Chapter 1 Prerequisites for Calculus Exercise 1.1 40E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 41E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 42E

    (d) As the best insulator will have the largest temperature change per inch as this will allow larger temperature changes on the other side of thin walls. Therefore, the best insulator is fiberglass insulation whereas the poorest is gypsum wallboard.

    Chapter 1 Prerequisites for Calculus Exercise 1.1 43E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 44E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 45E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 46E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 47E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 48E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 49E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 50E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 51E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 52E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 53E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 54E

    Chapter 1 Prerequisites for Calculus Exercise 1.1 55E
    The three points can be joined in three different ways to torm three different parallelograms as shown below:
    From the above three graphs we can see that the three missing pints are (5, 2), (-1, 4), and (-1, -2) respectively.

    Chapter 1 Prerequisites for Calculus Exercise 1.1 56E
    We nave to snow that it the midpoints ot consecutive sides ot any quadrilateral are connected, then the resulting figure is a parallelogram.

    From the above, the slopes of the four lines, we can see that two slopes each are of equal values. This means that those two lines are parallel to each other.
    Therefore from here we conclude that if the midpoints of consecutive sides of any quadrilateral are connected, then the resulting figure is a parallelogram.

    Chapter 1 Prerequisites for Calculus Exercise 1.1 57E
    As given in the question, the radius of the circle is 5, and the centre of the circle is at (0, 0) . The tangent passes through the point (3, 4).

    Chapter 1 Prerequisites for Calculus Exercise 1.1 58E



    10.10: Chapter 10 Practice

    In order to have a correlation coefficient between traits (A) and (B), it is necessary to have:

    1. one group of subjects, some of whom possess characteristics of trait (A), the remainder possessing those of trait (B)
    2. measures of trait (A) on one group of subjects and of trait (B) on another group two groups of subjects, one which could be classified as (A) or not (A), the other as (B) or not (B)
    3. two groups of subjects, one which could be classified as (A) or not (A), the other as (B) or not (B)

    Define the Correlation Coefficient and give a unique example of its use.

    If the correlation between age of an auto and money spent for repairs is +.90

    1. 81% of the variation in the money spent for repairs is explained by the age of the auto
    2. 81% of money spent for repairs is unexplained by the age of the auto 90% of the money spent for repairs is explained by the age of the auto
    3. none of the above

    Suppose that college grade-point average and verbal portion of an IQ test had a correlation of .40. What percentage of the variance do these two have in common?

    True or false? If false, explain why: The coefficient of determination can have values between -1 and +1.

    True or False: Whenever r is calculated on the basis of a sample, the value which we obtain for r is only an estimate of the true correlation coefficient which we would obtain if we calculated it for the entire population.

    Under a "scatter diagram" there is a notation that the coefficient of correlation is .10. What does this mean?

    1. plus and minus 10% from the means includes about 68% of the cases
    2. one-tenth of the variance of one variable is shared with the other variable one-tenth of one variable is caused by the other variable
    3. on a scale from -1 to +1, the degree of linear relationship between the two variables is +.10

    The correlation coefficient for (X) and (Y) is known to be zero. We then can conclude that:

    1. X and (Y) have standard distributions
    2. the variances of (X) and (Y) are equal
    3. there exists no relationship between (X) and Y there exists no linear relationship between (X) and Y
    4. none of these

    What would you guess the value of the correlation coefficient to be for the pair of variables: "number of hours worked" and "number of units of work completed"?

    1. Approximately 0.9
    2. Approximately 0.4
    3. Approximately 0.0 Approximately -0.4
    4. Approximately -0.9

    In a given group, the correlation between height measured in feet and weight measured in pounds is +.68. Which of the following would alter the value of r?

    1. height is expressed centimeters.
    2. weight is expressed in Kilograms.
    3. both of the above will affect r.
    4. neither of the above changes will affect r.

    10.2 Testing the Significance of the Correlation Coefficient

    Define a (t) Test of a Regression Coefficient, and give a unique example of its use.

    The correlation between scores on a neuroticism test and scores on an anxiety test is high and positive therefore

    1. anxiety causes neuroticism
    2. those who score low on one test tend to score high on the other.
    3. those who score low on one test tend to score low on the other. no prediction from one test to the other can be meaningfully made.

    10.3 Linear Equations

    True or False? If False, correct it: Suppose a 95% confidence interval for the slope (eta) of the straight line regression of (Y) on (X) is given by (-3.5 < eta < -0.5). Then a two-sided test of the hypothesis (H_ <0>: eta=-1) would result in rejection of (H_0) at the 1% level of significance.

    True or False: It is safer to interpret correlation coefficients as measures of association rather than causation because of the possibility of spurious correlation.

    We are interested in finding the linear relation between the number of widgets purchased at one time and the cost per widget. The following data has been obtained:

    (X): Number of widgets purchased &ndash 1, 3, 6, 10, 15

    (Y): Cost per widget(in dollars) &ndash 55, 52, 46, 32, 25

    Suppose the regression line is (hat=-2.5 x+60). We compute the average price per widget if 30 are purchased and observe which of the following?

    1. (hat=15 ext < dollars >) obviously, we are mistaken the prediction (hat y) is actually +15 dollars.
    2. (hat=15 ext < dollars >), which seems reasonable judging by the data.
    3. (hat=-15 ext < dollars >), which is obvious nonsense. The regression line must be incorrect.
    4. (hat=-15 ext < dollars >), which is obvious nonsense. This reminds us that predicting (Y) outside the range of (X) values in our data is a very poor practice.

    Discuss briefly the distinction between correlation and causality.

    True or False: If (r) is close to + or -1, we shall say there is a strong correlation, with the tacit understanding that we are referring to a linear relationship and nothing else.

    10.4 The Regression Equation

    Suppose that you have at your disposal the information below for each of 30 drivers. Propose a model (including a very brief indication of symbols used to represent independent variables) to explain how miles per gallon vary from driver to driver on the basis of the factors measured.

    1. miles driven per day
    2. weight of car
    3. number of cylinders in car
    4. average speed miles per gallon
    5. number of passengers

    Consider a sample least squares regression analysis between a dependent variable ((Y)) and an independent variable ((X)). A sample correlation coefficient of &minus1 (minus one) tells us that

    1. there is no relationship between (Y) and (X) in the sample
    2. there is no relationship between (Y) and (X) in the population there is a perfect negative relationship between (Y) and (X) in the population
    3. there is a perfect negative relationship between (Y) and (X) in the sample.

    In correlational analysis, when the points scatter widely about the regression line, this means that the correlation is

    10.5 Interpretation of Regression Coefficients: Elasticity and Logarithmic Transformation

    In a linear regression, why do we need to be concerned with the range of the independent ((X)) variable?

    Suppose one collected the following information where (X) is diameter of tree trunk and (Y) is tree height.

    X Y
    4 8
    2 4
    8 18
    6 22
    10 30
    6 8

    Table 10.3

    Regression equation: (hat_=-3.6+3.1 cdot X_)

    What is your estimate of the average height of all trees having a trunk diameter of 7 inches?

    The manufacturers of a chemical used in flea collars claim that under standard test conditions each additional unit of the chemical will bring about a reduction of 5 fleas (i.e. where (X_= ext < amount of chemical >) and(Y_=B_<0>+B_ <1>cdot X_+E_) ,(H_0:B_1=&minus5)

    Suppose that a test has been conducted and results from a computer include:

    Standard error of the regression coefficient = 1.0

    Degrees of Freedom for Error = 2000

    95% Confidence Interval for the slope &minus2.04, &minus5.96

    Is this evidence consistent with the claim that the number of fleas is reduced at a rate of 5 fleas per unit chemical?

    10.6 Predicting with a Regression Equation

    True or False? If False, correct it: Suppose you are performing a simple linear regression of (Y) on (X) and you test the hypothesis that the slope (eta) is zero against a two-sided alternative. You have (n=25) observations and your computed test ((t)) statistic is 2.6. Then your P-value is given by (.01 < P < .02), which gives borderline significance (i.e. you would reject (H_0) at (alpha=.02) but fail to reject (H_0) at (alpha=.01)).

    An economist is interested in the possible influence of "Miracle Wheat" on the average yield of wheat in a district. To do so he fits a linear regression of average yield per year against year after introduction of "Miracle Wheat" for a ten year period.

    ((Y_j): Average yield in (j) year after introduction)

    ((X_j): (j) year after introduction).

      What is the estimated average yield for the fourth year after introduction?
  • Do you want to use this trend line to estimate yield for, say, 20 years after introduction? Why? What would your estimate be?
  • An interpretation of (r=0.5) is that the following part of the (Y)-variation is associated with which variation in (X):

    Which of the following values of (r) indicates the most accurate prediction of one variable from another?

    10.7 How to Use Microsoft Excel® for Regression Analysis

    A computer program for multiple regression has been used to fit (hat_=b_<0>+b_ <1>cdot X_<1 j>+b_ <2>cdot X_<2 j>+b_ <3>cdot X_<3 j>).

    Part of the computer output includes:

    i (b_i) (S_)
    0 8 1.6
    1 2.2 .24
    2 -.72 .32
    3 0.005 0.002

    Table 10.4

    1. Calculation of confidence interval for (b_2) consists of _______(pm) (a student's (t) value) (_______) The confidence level for this interval is reflected in the value used for _______.
    2. The degrees of freedom available for estimating the variance are directly concerned with the value used for _______

    An investigator has used a multiple regression program on 20 data points to obtain a regression equation with 3 variables. Part of the computer output is:


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