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Calculus 1 : Rate

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Example Questions

Example Question #831 : Rate

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 10?

Possible Answers:

$160\pi$

$640\pi$

$320\pi$

$6400\pi$

$1600\pi$

Correct answer:

$640\pi$

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 10, we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (4)^2 (10)=640\pi$

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Example Question #743 : 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 15?

Possible Answers:

$2025\pi$

$8100\pi$

$135\pi$

$540\pi$

$2700\pi$

Correct answer:

$540\pi$

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 15, we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (3)^2 (15)=540\pi$

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Example Question #3651 : Calculus

A spherical balloon is being filled with air. What is the rate of growth of the sphere's volume when the radius is 2 and the rate of change of the radius is 14?

Possible Answers:

$112\pi$

$392\pi$

$224\pi$

$56\pi$

$448\pi$

Correct answer:

$112\pi$

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 2 and the rate of change of the radius is 14, we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (2)^2 (14)=112\pi$

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Example Question #3652 : Calculus

A spherical balloon is being filled with air. What is the rate of growth of the sphere's volume when the radius is 2 and the rate of change of the radius is 9?

Possible Answers:

$18\pi$

$648\pi$

$72\pi$

$162\pi$

$9\pi$

Correct answer:

$72\pi$

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 2 and the rate of change of the radius is 9, we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (2)^2 (9)=72\pi$

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Example Question #743 : 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 1 and the rate of change of the radius is 31?

Possible Answers:

$124\pi$

$620\pi$

$3844\pi$

$961\pi$

$31\pi$

Correct answer:

$124\pi$

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 1 and the rate of change of the radius is 31, we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (1)^2 (31)=124\pi$

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Example Question #743 : 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 2 and the rate of change of the radius is 23?

Possible Answers:

$92\pi$

$184\pi$

$529\pi$

$46\pi$

$2116\pi$

Correct answer:

$184\pi$

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 2 and the rate of change of the radius is 23, we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (2)^2 (23)=184\pi$

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Example Question #741 : 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 2 and the rate of change of the radius is 21?

Possible Answers:

$42\pi$

$441\pi$

$1764\pi$

$882\pi$

$168\pi$

Correct answer:

$168\pi$

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 2 and the rate of change of the radius is 21, we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (2)^2 (21)=168\pi$

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Example Question #745 : 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 23?

Possible Answers:

$276\pi$

$3174\pi$

$207\pi$

$529\pi$

$828\pi$

Correct answer:

$828\pi$

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 23, we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (3)^2 (23)=828\pi$

Report an Error

Example Question #743 : 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 21?

Possible Answers:

$7056\pi$

$588\pi$

$336\pi$

$1344\pi$

$1764\pi$

Correct answer:

$1344\pi$

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 21, we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (4)^2 (21)=1344\pi$

Report an Error

Example Question #751 : 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 9?

Possible Answers:

$72\pi$

$144\pi$

$576\pi$

$16\pi$

$288\pi$

Correct answer:

$144\pi$

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 9, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(2)(9)=144\pi$

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Calculus 1 : Rate

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #831 : Rate

Example Question #743 : Rate Of Change

Example Question #3651 : Calculus

Example Question #3652 : Calculus

Example Question #743 : How To Find Rate Of Change

Example Question #743 : Rate Of Change

Example Question #741 : Rate Of Change

Example Question #745 : Rate Of Change

Example Question #743 : Rate Of Change

Example Question #751 : How To Find Rate Of Change

All Calculus 1 Resources

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