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Calculus 1 : Rate of Change

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Example Questions

Example Question #391 : Rate

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:

$\frac{\pi}{4}$

$4\pi$

$\frac{\pi}{2}$

$2\pi$

$\pi$

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 #311 : How To Find 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

289

578

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 #312 : How To Find 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:

49.30

69.45

37.21

51.27

46.15

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 #313 : How To Find 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

50.4

6.3

25.2

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 #314 : How To Find 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:

$32\pi^2$

$8\pi$

$16\pi^2$

$32\pi$

$16\pi$

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 #315 : How To Find 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:

$12\pi$

$3\pi$

$12\pi$

$4\pi$

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 #316 : How To Find 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:

18.12

15.56

19.45

11.67

12.09

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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Example Question #317 : How To Find Rate Of Change

Differentiate the following function:

$f(x)=e^{x^2+cosx}$

Possible Answers:

$(2x+cosx)e^{x^{2}-sinx}$

$(2x+sinx)e^{x}$

$e^{2x-sinx}$

$(2x-sinx)e^{x^{2}+cosx}$

$(2x+cosx)e^{x}$

Correct answer:

$(2x-sinx)e^{x^{2}+cosx}$

Explanation:

To take the derivative of a composite function of the form f(x)=f(g(x)) one must use the chain rule which states: f'(x)=g'(x)f'(g(x))

The derivative of the 'inner function' for this question would be 2x-Sinx while the outer function stays the same since the derivative of $e^{x}$ is itself.

From this we find that the derivative is

$(2x-Sinx)e^{x^2+Cosx}$

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Example Question #318 : How To Find Rate Of Change

A cube is growing in size. What is the length of the sides of the cube at the time that the rate of growth of the cube's volume is equal to 0.56 times the rate of growth of its surface area?

Possible Answers:

5.60

2.24

1.12

3.36

4.48

Correct answer:

2.24

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its volume and surface area in terms of the length of its sides:

V=s^3

A=6s^2

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

$\frac{dA}{dt}=12s\frac{ds}{dt}$

Now, solve for the length of the side of the cube to satisfy the problem condition, the rate of growth of the cube's volume is equal to 0.56 times the rate of growth of its surface area:

$3s^2\frac{ds}{dt}=(0.56)12s\frac{ds}{dt}$

s=(0.56)4=2.24

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Example Question #319 : How To Find Rate Of Change

A cube is diminishing in size. What is the length of the diagonal of the cube at the time that the rate of shrinkage of the cube's volume is equal to 0.34 times the rate of shrinkage of its surface area?

Possible Answers:

1.97

1.36

2.80

2.36

4.12

Correct answer:

2.36

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its volume and surface area in terms of the length of its sides:

V=s^3

A=6s^2

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

$\frac{dA}{dt}=12s\frac{ds}{dt}$

Now, solve for the length of the side of the cube to satisfy the problem condition, the rate of shrinkage of the cube's volume is equal to 0.34 times the rate of shrinkage of its surface area:

$3s^2\frac{ds}{dt}=(0.34)12s\frac{ds}{dt}$

s=(0.34)4=1.36

The diagonal is then found from this:

$d=s\sqrt{3}$

$d=1.36\sqrt{3}=2.36$

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Calculus 1 : Rate of Change

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #391 : Rate

Example Question #311 : How To Find Rate Of Change

Example Question #312 : How To Find Rate Of Change

Example Question #313 : How To Find Rate Of Change

Example Question #314 : How To Find Rate Of Change

Example Question #315 : How To Find Rate Of Change

Example Question #316 : How To Find Rate Of Change

Example Question #317 : How To Find Rate Of Change

Example Question #318 : How To Find Rate Of Change

Example Question #319 : How To Find Rate Of Change

All Calculus 1 Resources

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