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Calculus 1 : Rate of Change

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Example Questions

Example Question #661 : Rate Of Change

A cube is growing in size. What is the ratio of the rate of growth of the cube's volume to the rate of growth of its sides when its sides have length 30?

Possible Answers:

5400

2700

8100

900

Correct answer:

2700

Explanation:

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

V=s^3

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the volume and sides:

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$

$\phi =3(30)^2=2700$

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Example Question #662 : How To Find Rate Of Change

A cube is growing in size. What is the ratio of the rate of growth of the cube's volume to the rate of growth of its sides when its sides have length 32?

Possible Answers:

27648

3072

9216

1024

Correct answer:

3072

Explanation:

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

V=s^3

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the volume and sides:

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$

$\phi =3(32)^2=3072$

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Example Question #661 : How To Find Rate Of Change

A cube is growing in size. What is the ratio of the rate of growth of the cube's volume to the rate of growth of its sides when its sides have length 95?

Possible Answers:

285

81225

855

9025

27075

Correct answer:

27075

Explanation:

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

V=s^3

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the volume and sides:

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$

$\phi =3(95)^2=27075$

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Example Question #662 : How To Find Rate Of Change

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's volume to the rate of loss of its sides when its sides have length $\frac{14}{15}$ ?

Possible Answers:

$\frac{14}{75}$

$\frac{196}{225}$

$\frac{196}{75}$

$\frac{196}{25}$

$\frac{14}{225}$

Correct answer:

$\frac{196}{75}$

Explanation:

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

V=s^3

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the volume and sides:

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$

$\phi =3(\frac{14}{15})^2=\frac{196}{75}$

Report an Error

Example Question #665 : How To Find Rate Of Change

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's volume to the rate of loss of its sides when its sides have length $\frac{13}{9}$ ?

Possible Answers:

$\frac{13}{27}$

$\frac{169}{81}$

$\frac{169}{27}$

$\frac{13}{81}$

$\frac{169}{9}$

Correct answer:

$\frac{169}{27}$

Explanation:

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

V=s^3

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the volume and sides:

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$

$\phi =3(\frac{13}{9})^2=\frac{169}{27}$

Report an Error

Example Question #666 : How To Find Rate Of Change

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's volume to the rate of loss of its sides when its sides have length $\frac{11}{4}$ ?

Possible Answers:

$\frac{363}{16}$

$\frac{363}{4}$

$\frac{121}{4}$

$\frac{121}{8}$

$\frac{121}{16}$

Correct answer:

$\frac{363}{16}$

Explanation:

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

V=s^3

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the volume and sides:

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$

$\phi =3(\frac{11}{4})^2=\frac{363}{16}$

Report an Error

Example Question #3575 : Calculus

A cube is growing in size. What is the ratio of the rate of growth of the cube's surface area to the rate of growth of its diagonal when its sides have length $11\sqrt{17}$ ?

Possible Answers:

$44\sqrt{51}$

$132\sqrt{3}$

$17\sqrt{51}$

$132\sqrt{17}$

$33\sqrt{51}$

Correct answer:

$44\sqrt{51}$

Explanation:

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

A=6s^2

$d=s\sqrt{3}$

The rates of change of these can be found by taking the derivative of each side of the equations with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the surface area and diagonal:

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(11\sqrt{17})\sqrt{3}=44\sqrt{51}$

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Example Question #2551 : Functions

A cube is growing in size. What is the ratio of the rate of growth of the cube's surface area to the rate of growth of its diagonal when its sides have length $2\sqrt{10}$ ?

Possible Answers:

$4\sqrt{15}$

$8\sqrt{30}$

$80\sqrt{3}$

$12\sqrt{15}$

$16\sqrt{15}$

Correct answer:

$8\sqrt{30}$

Explanation:

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

A=6s^2

$d=s\sqrt{3}$

The rates of change of these can be found by taking the derivative of each side of the equations with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the surface area and diagonal:

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(2\sqrt{10})\sqrt{3}=8\sqrt{30}$

Report an Error

Example Question #669 : How To Find Rate Of Change

A cube is growing in size. What is the ratio of the rate of growth of the cube's surface area to the rate of growth of its diagonal when its sides have length $2\sqrt{42}$ ?

Possible Answers:

$8\sqrt{42}$

$12\sqrt{14}$

$24\sqrt{14}$

$16\sqrt{42}$

$7\sqrt{51}$

Correct answer:

$24\sqrt{14}$

Explanation:

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

A=6s^2

$d=s\sqrt{3}$

The rates of change of these can be found by taking the derivative of each side of the equations with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the surface area and diagonal:

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(2\sqrt{42})\sqrt{3}=24\sqrt{14}$

Report an Error

Example Question #670 : How To Find Rate Of Change

A cube is growing in size. What is the ratio of the rate of growth of the cube's surface area to the rate of growth of its diagonal when its sides have length $3\sqrt{39}$ ?

Possible Answers:

$36\sqrt{39}$

$12\sqrt{13}$

$36\sqrt{13}$

$12\sqrt{39}$

$108\sqrt{13}$

Correct answer:

$36\sqrt{13}$

Explanation:

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

A=6s^2

$d=s\sqrt{3}$

The rates of change of these can be found by taking the derivative of each side of the equations with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the surface area and diagonal:

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(3\sqrt{39})\sqrt{3}=36\sqrt{13}$

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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 #661 : Rate Of Change

Example Question #662 : How To Find Rate Of Change

Example Question #661 : How To Find Rate Of Change

Example Question #662 : How To Find Rate Of Change

Example Question #665 : How To Find Rate Of Change

Example Question #666 : How To Find Rate Of Change

Example Question #3575 : Calculus

Example Question #2551 : Functions

Example Question #669 : How To Find Rate Of Change

Example Question #670 : How To Find Rate Of Change

All Calculus 1 Resources

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