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Calculus 1 : Calculus

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Example Questions

Example Question #756 : 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 23?

Possible Answers:

$736\pi$

$368\pi$

$46\pi$

$16928\pi$

$8464\pi$

Correct answer:

$368\pi$

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

$\frac{dA}{dt}=8\pi(2)(23)=368\pi$

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Example Question #757 : 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 21 and the rate of change of the radius is 2?

Possible Answers:

$21\pi$

$168\pi$

$42\pi$

$336\pi$

$84\pi$

Correct answer:

$336\pi$

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 21 and the rate of change of the radius is 2, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(21)(2)=336\pi$

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Example Question #758 : 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 4 and the rate of change of the radius is 10?

Possible Answers:

$40\pi$

$1280\pi$

$1600\pi$

$320\pi$

$3200\pi$

Correct answer:

$320\pi$

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 4 and the rate of change of the radius is 10, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(4)10()=320\pi$

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Example Question #761 : 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 14?

Possible Answers:

$28\pi$

$3136\pi$

$56\pi$

$224\pi$

$392\pi$

Correct answer:

$224\pi$

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

$\frac{dA}{dt}=8\pi(2)(14)=224\pi$

Report an Error

Example Question #762 : 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 31?

Possible Answers:

$248\pi$

$62\pi$

$7688\pi$

$961\pi$

$31\pi$

Correct answer:

$248\pi$

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

$\frac{dA}{dt}=8\pi(1)(31)=248\pi$

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Example Question #763 : How To Find Rate Of Change

A cube is growing in size. What is the rate of growth of one of the cube's faces if its sides have a length of 4 and a rate of growth of 21?

Possible Answers:

147

441

168

336

Correct answer:

168

Explanation:

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

a=s^2

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

$\frac{da}{dt}=2s\frac{ds}{dt}$

Once we have the rate equation for the area of the face, we can use what we know about the cube, specifically that its sides have a length of 4 and a rate of growth of 21:

$\frac{da}{dt}=2(4)(21)=168$

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Example Question #764 : How To Find Rate Of Change

A cube is growing in size. What is the rate of growth of one of the cube's faces if its sides have a length of 2 and a rate of growth of 21?

Possible Answers:

441

7056

3528

504

Correct answer:

Explanation:

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

a=s^2

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

$\frac{da}{dt}=2s\frac{ds}{dt}$

Once we have the rate equation for the area of the face, we can use what we know about the cube, specifically that its sides have a length of 2 and a rate of growth of 21:

$\frac{da}{dt}=2(2)(21)=84$

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Example Question #761 : How To Find Rate Of Change

A cube is growing in size. What is the rate of growth of one of the cube's faces if its sides have a length of 23 and a rate of growth of 4?

Possible Answers:

2116

184

Correct answer:

184

Explanation:

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

a=s^2

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

$\frac{da}{dt}=2s\frac{ds}{dt}$

Once we have the rate equation for the area of the face, we can use what we know about the cube, specifically that its sides have a length of 23 and a rate of growth of 4:

$\frac{da}{dt}=2(23)(4)=184$

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Example Question #762 : How To Find Rate Of Change

A cube is growing in size. What is the rate of growth of one of the cube's faces if its sides have a length of 20 and a rate of growth of 1?

Possible Answers:

400

800

160

Correct answer:

Explanation:

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

a=s^2

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

$\frac{da}{dt}=2s\frac{ds}{dt}$

Once we have the rate equation for the area of the face, we can use what we know about the cube, specifically that its sides have a length of 20 and a rate of growth of 1:

$\frac{da}{dt}=2(20)(1)=40$

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Example Question #763 : How To Find Rate Of Change

A cube is growing in size. What is the rate of growth of one of the cube's faces if its sides have a length of 3 and a rate of growth of 15?

Possible Answers:

2025

270

4050

Correct answer:

Explanation:

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

a=s^2

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

$\frac{da}{dt}=2s\frac{ds}{dt}$

Once we have the rate equation for the area of the face, we can use what we know about the cube, specifically that its sides have a length of 3 and a rate of growth of 15:

$\frac{da}{dt}=2(3)(15)=90$

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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #756 : How To Find Rate Of Change

Example Question #757 : How To Find Rate Of Change

Example Question #758 : How To Find Rate Of Change

Example Question #761 : How To Find Rate Of Change

Example Question #762 : How To Find Rate Of Change

Example Question #763 : How To Find Rate Of Change

Example Question #764 : How To Find Rate Of Change

Example Question #761 : How To Find Rate Of Change

Example Question #762 : How To Find Rate Of Change

Example Question #763 : How To Find Rate Of Change

All Calculus 1 Resources

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