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Calculus 1 : Calculus

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10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #301 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the volume of the sphere at the instance the rate of growth of the volume is 1.62 times the rate of growth of the circumference?

Possible Answers:

1.009

3.054

1.527

2.018

4.036

Correct answer:

3.054

Explanation:

Begin by writing the equations for the volume and circumference of a sphere with respect to the sphere's radius:

$V=\frac{4}{3}\pi r^3$

$C=2\pi r^$

The rates of change can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. Now given the problem information, the rate of growth of the volume is 1.62 times the rate of growth of the circumference, solve for the radius:

$4\pi r^2 \frac{dr}{dt}=(1.62)2\pi \frac{dr}{dt}$

$r^2=(1.62)\frac{1}{2}$

$r =\sqrt{\frac{1.62}{2}}$

r=.9

To find the volume:

$V=\frac{4}{3}\pi r^3$

$V=\frac{4}{3}\pi (0.9)^3=3.054$

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Example Question #302 : How To Find Rate Of Change

A spherical balloon is deflating, while maintaining its spherical shape. What is the surface area of the sphere at the instance the rate of shrinkage of the volume is 3.24 times the rate of shrinkage of the circumference?

Possible Answers:

6.21

24.82

20.36

5.09

12.41

Correct answer:

20.36

Explanation:

Begin by writing the equations for the volume and circumference of a sphere with respect to the sphere's radius:

$V=\frac{4}{3}\pi r^3$

$C=2\pi r^$

The rates of change can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. Now given the problem information, the rate of shrinkage of the volume is 3.24 times the rate of shrinkage of the circumference, solve for the radius:

$4\pi r^2 \frac{dr}{dt}=(3.24)2\pi \frac{dr}{dt}$

$r^2=(3.24)\frac{1}{2}$

$r =\sqrt{\frac{3.24}{2}}$

$r=0.9\sqrt{2}$

Then, to find the surface area:

$A=4\pi r^2$

$A=4\pi (0.9\sqrt{2})^2=20.36$

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Example Question #303 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the circumference of the sphere at the instance the rate of growth of the surface area is 9.96 times the rate of growth of the circumference?

Possible Answers:

20.29

5.81

15.65

17.43

2.49

Correct answer:

15.65

Explanation:

Start by writing the equations for the surface area and circumference of a sphere with respect to the sphere's radius:

$A=4\pi r^2$

$C=2\pi r$

The rates of change can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dA}{dt}=8\pi r \frac{dr}{dt}$

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. Now we can use the relation given in the problem statement, , to solve for the length of the radius at that instant:

$8\pi r \frac{dr}{dt}=(9.96)2\pi \frac{dr}{dt}$

$r =\frac{9.96}{4}$

r=2.49

And finally to find the circumference:

$C=2\pi r$

$C=2\pi (2.49)=15.65$

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Example Question #311 : Rate Of Change

A spherical balloon is being filled with air. What is ratio of the rate of growth of the volume of the sphere to the rate of growth of the surface area when the radius is $2\pi$ ?

Possible Answers:

$\pi$

$4\pi$

$\frac{\pi}{4}$

$\frac{\pi}{2}$

$2\pi$

Correct answer:

$\pi$

Explanation:

Let's begin by writing the equations for the volume and surface area of a sphere with respect to the sphere's radius:

$V=\frac{4}{3}\pi r^3$

$A=4\pi r^2$

The rates of change can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$

$\frac{dA}{dt}=8\pi r \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. To find the ratio of the rates of changes of the volume and surface area, divide:

$4\pi r^2 \frac{dr}{dt}=(\phi)8\pi r \frac{dr}{dt}$

$\phi =\frac{r}{2}$

$\phi =\frac{2\pi}{2}=\pi$

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Example Question #312 : Rate Of Change

A spherical balloon is being filled with air. What is ratio of the rate of growth of the volume of the sphere to the rate of growth of the circumference when the radius is 17?

Possible Answers:

476

578

289

Correct answer:

578

Explanation:

Let's begin by writing the equations for the volume and circumference of a sphere with respect to the sphere's radius:

$V=\frac{4}{3}\pi r^3$

$C=2\pi r^$

The rates of change can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. To find the ratio of the rates of changes of the volume and circumference, divide:

$4\pi r^2 \frac{dr}{dt}=(\phi)2\pi \frac{dr}{dt}$

$\phi =2r^2$

$\phi =2(17)^2=578$

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Example Question #313 : Rate Of Change

A spherical balloon is deflating, although it maintains its spherical shape. What is ratio of the rate of loss of the volume of the sphere to the rate of loss of the surface area when the radius is 92.3?

Possible Answers:

46.15

37.21

51.27

49.30

69.45

Correct answer:

46.15

Explanation:

Let's begin by writing the equations for the volume and surface area of a sphere with respect to the sphere's radius:

$V=\frac{4}{3}\pi r^3$

$A=4\pi r^2$

The rates of change can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$

$\frac{dA}{dt}=8\pi r \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. To find the ratio of the rates of changes of the volume and surface area, divide:

$4\pi r^2 \frac{dr}{dt}=(\phi)8\pi r \frac{dr}{dt}$

$\phi =\frac{r}{2}$

$\phi =\frac{92.3}{2}$

$\phi = 46.15$

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Example Question #314 : Rate Of Change

A spherical balloon is deflating, while maintaining its spherical shape. What is the radius of the sphere at the instance the rate of shrinkage of the surface area is 25.2 times the rate of shrinkage of the circumference?

Possible Answers:

12.6

8.4

25.2

6.3

50.4

Correct answer:

6.3

Explanation:

Start by writing the equations for the surface area and circumference of a sphere with respect to the sphere's radius:

$A=4\pi r^2$

$C=2\pi r$

The rates of change can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dA}{dt}=8\pi r \frac{dr}{dt}$

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. Now we can use the relation given in the problem statement, the rate of growth of the surface area is 25.2 times the rate of growth of the circumference, to solve for the length of the radius at that instant:

$8\pi r \frac{dr}{dt}=(25.2)2\pi \frac{dr}{dt}$

$r =\frac{25.2}{4}$

r=6.3

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Example Question #315 : Rate Of Change

A spherical balloon is deflating, although it retains a spherical shape. What is ratio of the rate of loss of the volume of the sphere to the rate of loss of the circumference when the radius is $4\pi$ ?

Possible Answers:

$32\pi^2$

$8\pi$

$32\pi$

$16\pi$

$16\pi^2$

Correct answer:

$32\pi^2$

Explanation:

Let's begin by writing the equations for the volume and circumference of a sphere with respect to the sphere's radius:

$V=\frac{4}{3}\pi r^3$

$C=2\pi r^$

The rates of change can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. To find the ratio of the rates of changes of the volume and circumference, divide:

$4\pi r^2 \frac{dr}{dt}=(\phi)2\pi \frac{dr}{dt}$

$\phi =2r^2$

$\phi =2(4\pi)^2=32\pi^2$

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Example Question #316 : Rate Of Change

A spherical balloon is being filled with air. What is ratio of the rate of growth of the surface area of the sphere to the rate of growth of the circumference when the radius is $3\pi$ ?

Possible Answers:

$3\pi$

$12\pi$

$4\pi$

$12\pi$

Correct answer:

$12\pi$

Explanation:

Let's begin by writing the equations for the surface area and circumference of a sphere with respect to the sphere's radius:

$A=4\pi r^2$

$C=2\pi r$

The rates of change can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dA}{dt}=8\pi r \frac{dr}{dt}$

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. To find the ratio of the rates of changes of the surface area and circumference, divide:

$8\pi r \frac{dr}{dt}=(\phi)2\pi \frac{dr}{dt}$

$\phi =4r$

$\phi =4(3\pi)=12\pi$

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Example Question #317 : Rate Of Change

A spherical balloon is deflating, although it retains a spherical shape. What is ratio of the rate of loss of the surface area of the sphere to the rate of loss of the circumference when the radius is 3.89?

Possible Answers:

15.56

12.09

19.45

11.67

18.12

Correct answer:

15.56

Explanation:

Let's begin by writing the equations for the surface area and circumference of a sphere with respect to the sphere's radius:

$A=4\pi r^2$

$C=2\pi r$

The rates of change can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dA}{dt}=8\pi r \frac{dr}{dt}$

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

$8\pi r \frac{dr}{dt}=(\phi)2\pi \frac{dr}{dt}$

$\phi =4r$

$\phi =4(3.89)=15.56$

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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #301 : How To Find Rate Of Change

Example Question #302 : How To Find Rate Of Change

Example Question #303 : How To Find Rate Of Change

Example Question #311 : Rate Of Change

Example Question #312 : Rate Of Change

Example Question #313 : Rate Of Change

Example Question #314 : Rate Of Change

Example Question #315 : Rate Of Change

Example Question #316 : Rate Of Change

Example Question #317 : Rate Of Change

All Calculus 1 Resources

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