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Calculus 1 : Rate of Change

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Example Questions

Example Question #721 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's volume when the radius is 3 and the rate of change of the radius is $\frac{1}{2}$ ?

Possible Answers:

$9\pi$

$36\pi$

$18\pi$

$54\pi$

$27\pi$

Correct answer:

$18\pi$

Explanation:

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

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

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

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

Now given our problem conditions, the radius is 3 and the rate of change of the radius is $\frac{1}{2}$ , we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (3)^2 (\frac{1}{2})=18\pi$

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Example Question #722 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's volume when the radius is 5 and the rate of change of the radius is 5?

Possible Answers:

$2500\pi$

$500\pi$

$1250\pi$

$\frac{2500\pi}{3}$

$250\pi$

Correct answer:

$500\pi$

Explanation:

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

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

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

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

Now given our problem conditions, the radius is 5 and the rate of change of the radius is 5, we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (5)^2 (5)=500\pi$

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Example Question #723 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's volume when the radius is 9 and the rate of change of the radius is 1?

Possible Answers:

$216\pi$

$108\pi$

$972\pi$

$324\pi$

$729\pi$

Correct answer:

$324\pi$

Explanation:

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

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

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

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

Now given our problem conditions, the radius is 9 and the rate of change of the radius is 1, we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (9)^2 (1)=324\pi$

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Example Question #724 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's volume when the radius is 11 and the rate of change of the radius is 3?

Possible Answers:

$132\pi$

$1452\pi$

$484\pi$

$363\pi$

$5324\pi$

Correct answer:

$1452\pi$

Explanation:

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

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

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

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

Now given our problem conditions, the radius is 11 and the rate of change of the radius is 3, we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (11)^2 (3)=1452\pi$

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Example Question #725 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's volume when the radius is 4 and the rate of change of the radius is $\frac{1}{8}$ ?

Possible Answers:

$8\pi$

$\frac{32\pi}{3}$

$\pi$

$2\pi$

$32\pi$

Correct answer:

$8\pi$

Explanation:

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

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

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

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

Now given our problem conditions, the radius is 4 and the rate of change of the radius is $\frac{1}{8}$ , we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (4)^2 (\frac{1}{8})=8\pi$

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Example Question #726 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's volume when the radius is 5 and the rate of change of the radius is 11?

Possible Answers:

$5500\pi$

$550\pi$

$\frac{5500\pi}{3}$

$\frac{1100\pi}{3}$

$1100\pi$

Correct answer:

$1100\pi$

Explanation:

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

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

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

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

Now given our problem conditions, the radius is 5 and the rate of change of the radius is 11, we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (5)^2 (11)=1100\pi$

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Example Question #727 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's surface area when the radius is 1 and the rate of change of the radius is 6?

Possible Answers:

$192\pi$

$96\pi$

$48\pi$

$12\pi$

$6\pi$

Correct answer:

$48\pi$

Explanation:

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

$A=4\pi r^2$

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

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

Considering what was given as our problem conditions, the radius is 1 and the rate of change of the radius is 6, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi (1)(6)=48\pi$

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Example Question #728 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's surface area when the radius is 2 and the rate of change of the radius is 4?

Possible Answers:

$8\pi$

$64\pi$

$16\pi$

$32\pi$

$4\pi$

Correct answer:

$64\pi$

Explanation:

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

$A=4\pi r^2$

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

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

Considering what was given as our problem conditions, the radius is 2 and the rate of change of the radius is 4, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(2)(4)=64\pi$

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Example Question #728 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's surface area when the radius is 3 and the rate of change of the radius is 12?

Possible Answers:

$144\pi$

$72\pi$

$36\pi$

$96\pi$

$288\pi$

Correct answer:

$288\pi$

Explanation:

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

$A=4\pi r^2$

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

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

Considering what was given as our problem conditions, the radius is 3 and the rate of change of the radius is 12, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(3)(12)=288\pi$

Report an Error

Example Question #729 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's surface area when the radius is 5 and the rate of change of the radius is 2?

Possible Answers:

$10\pi$

$160\pi$

$80\pi$

$40\pi$

$320\pi$

Correct answer:

$80\pi$

Explanation:

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

$A=4\pi r^2$

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

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

Considering what was given as our problem conditions, the radius is 5 and the rate of change of the radius is 2, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(5)(2)=80\pi$

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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 #721 : How To Find Rate Of Change

Example Question #722 : How To Find Rate Of Change

Example Question #723 : How To Find Rate Of Change

Example Question #724 : How To Find Rate Of Change

Example Question #725 : How To Find Rate Of Change

Example Question #726 : How To Find Rate Of Change

Example Question #727 : How To Find Rate Of Change

Example Question #728 : How To Find Rate Of Change

Example Question #728 : How To Find Rate Of Change

Example Question #729 : How To Find Rate Of Change

All Calculus 1 Resources

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