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Calculus 1 : Calculus

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Example Questions

Example Question #731 : 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 8 and the rate of change of the radius is 8?

Possible Answers:

$1024\pi$

$512\pi$

$8192\pi$

$64\pi$

$2048\pi$

Correct answer:

$512\pi$

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 8 and the rate of change of the radius is 8, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(8)(8)=512\pi$

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Example Question #734 : 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 23 and the rate of change of the radius is 4?

Possible Answers:

$4232\pi$

$2116\pi$

$16928\pi$

$1058\pi$

$8464\pi$

Correct answer:

$8464\pi$

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 23 and the rate of change of the radius is 4, we can solve for the rate of change of our volume:

$\frac{dV}{dt}=4\pi (23)^2 (4)=8464\pi$

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Example Question #735 : 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 21 and the rate of change of the radius is 2?

Possible Answers:

$3528\pi$

$882\pi$

$336\pi$

$84\pi$

$441\pi$

Correct answer:

$3528\pi$

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

$\frac{dV}{dt}=4\pi (21)^2 (2)=3528\pi$

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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 20 and the rate of change of the radius is 1?

Possible Answers:

$1600\pi$

$160\pi$

$400\pi$

$80\pi$

$40\pi$

Correct answer:

$1600\pi$

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

$\frac{dV}{dt}=4\pi (20)^2 (1)=1600\pi$

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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 #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 2 and the rate of change of the radius is 14?

Possible Answers:

$56\pi$

$224\pi$

$112\pi$

$392\pi$

$448\pi$

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 #742 : 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 2 and the rate of change of the radius is 9?

Possible Answers:

$648\pi$

$162\pi$

$72\pi$

$18\pi$

$9\pi$

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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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #731 : Rate Of Change

Example Question #734 : Rate Of Change

Example Question #735 : Rate Of Change

Example Question #741 : Rate Of Change

Example Question #831 : Rate

Example Question #743 : Rate Of Change

Example Question #831 : Rate

Example Question #742 : How To Find Rate Of Change

Example Question #743 : How To Find Rate Of Change

Example Question #743 : Rate Of Change

All Calculus 1 Resources

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