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Calculus 1 : Rate of Change

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Example Questions

Example Question #451 : 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 surface area when its sides have length 124?

Possible Answers:

164

Correct answer:

Explanation:

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

V=s^3

A=6s^2

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

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

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

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

$3s^2\frac{ds}{dt}=(\phi)12s\frac{ds}{dt}$

$\phi=\frac{s}{4}$

$\phi =\frac{124}{4}=31$

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Example Question #452 : 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 surface area when its sides have length 720?

Possible Answers:

360

180

240

120

Correct answer:

180

Explanation:

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

V=s^3

A=6s^2

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

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

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

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

$3s^2\frac{ds}{dt}=(\phi)12s\frac{ds}{dt}$

$\phi=\frac{s}{4}$

$\phi =\frac{720}{4}=180$

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Example Question #453 : 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 5?

Possible Answers:

225

150

Correct answer:

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 volumecan 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:

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

$\phi=3s^2$

$\phi =3(5)^2=75$

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Example Question #454 : 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 11?

Possible Answers:

363

121

1089

Correct answer:

363

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 volumecan 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:

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

$\phi=3s^2$

$\phi =3(11)^2=363$

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Example Question #455 : 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 $\frac{1}{3}$ ?

Possible Answers:

$\frac{1}{27}$

$\frac{1}{9}$

$\frac{1}{3}$

Correct answer:

$\frac{1}{3}$

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 volumecan 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:

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

$\phi=3s^2$

$\phi =3(\frac{1}{3})^2=\frac{1}{3}$

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Example Question #456 : 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 $\frac{1}{6}$ ?

Possible Answers:

$\frac{1}{8}$

$\frac{1}{24}$

$\frac{1}{6}$

$\frac{1}{12}$

$\frac{1}{4}$

Correct answer:

$\frac{1}{12}$

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 volumecan 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:

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

$\phi=3s^2$

$\phi =3(\frac{1}{6})^2=\frac{1}{12}$

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Example Question #457 : 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 7?

Possible Answers:

441

147

Correct answer:

147

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 volumecan 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:

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

$\phi=3s^2$

$\phi =3(7)^2=147$

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Example Question #458 : 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 $\frac{4}{3}$ ?

Possible Answers:

$\frac{16}{3}$

$\frac{16}{9}$

Correct answer:

$\frac{16}{3}$

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 volumecan 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:

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

$\phi=3s^2$

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

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Example Question #459 : How To Find Rate Of Change

Find the slope at x=1 given the following function: $f(x)= 4e^{2x} + 3x^4$

Possible Answers:

Answer not listed

71.1

8.1

54.5

29.7

Correct answer:

71.1

Explanation:

In order to find the slope of a certain point, you first find the derivative of the following function: $f(x)= 4e^{2x} + 3x^4$

The derivative is: $f'(x)= 8e^{2x} + 12x^{3}$

Then, plug x= 1 into the derivative: $f'(1)= 8e^{2({\color{Blue} 1})} + 12({\color{Blue} 1})^{3}$

Therefore, the slope is: 71.1

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Example Question #460 : How To Find Rate Of Change

Find the slope at x=0 given the following function: f(x)= ln(2x+e^x)

Possible Answers:

Answer not listed

Correct answer:

Explanation:

In order to find the slope of a certain point, you first find the derivative of the following function: f(x)= ln(2x+e^x)

The derivative is: $f'(x)= \frac{2+e^x}{2x+e^x}$

Then, plug x= 0 into the derivative: $f'(0)= \frac{2+e^{\color{Blue} 0}}{2({\color{Blue} 0})+e^{\color{Blue} 0}}$

Therefore, the slope is:

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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 #451 : How To Find Rate Of Change

Example Question #452 : How To Find Rate Of Change

Example Question #453 : How To Find Rate Of Change

Example Question #454 : How To Find Rate Of Change

Example Question #455 : How To Find Rate Of Change

Example Question #456 : How To Find Rate Of Change

Example Question #457 : How To Find Rate Of Change

Example Question #458 : How To Find Rate Of Change

Example Question #459 : How To Find Rate Of Change

Example Question #460 : How To Find Rate Of Change

All Calculus 1 Resources

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