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Calculus 1 : Rate of Change

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Example Questions

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:

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

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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 #765 : 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 #766 : 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:

800

400

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 #767 : 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:

4050

2025

270

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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Example Question #768 : 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 9?

Possible Answers:

324

648

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 9:

$\frac{da}{dt}=2(2)(9)=36$

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Example Question #769 : 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 1 and a rate of growth of 31?

Possible Answers:

124

248

1922

961

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 1 and a rate of growth of 31:

$\frac{da}{dt}=2(1)(31)=62$

Report an Error

Example Question #770 : 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 14?

Possible Answers:

112

784

392

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 14:

$\frac{da}{dt}=2(2)(14)=56$

Report an Error

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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 #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 #765 : How To Find Rate Of Change

Example Question #766 : How To Find Rate Of Change

Example Question #767 : How To Find Rate Of Change

Example Question #768 : How To Find Rate Of Change

Example Question #769 : How To Find Rate Of Change

Example Question #770 : How To Find Rate Of Change

All Calculus 1 Resources

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