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Calculus 1 : Rate

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Example Questions

Example Question #771 : Rate

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 sides when its sides have length 102?

Possible Answers:

2448

1224

204

612

1632

Correct answer:

1224

Explanation:

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

A=6s^2

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

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

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

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

$\phi=12s$

$\phi =12(102)=1224$

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Example Question #2561 : 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 sides when its sides have length 111?

Possible Answers:

666

444

2664

888

1332

Correct answer:

1332

Explanation:

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

A=6s^2

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

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

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

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

$\phi=12s$

$\phi =12(111)=1332$

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Example Question #2561 : 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 sides when its sides have length 205?

Possible Answers:

1080

2460

810

270

Correct answer:

2460

Explanation:

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

A=6s^2

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

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

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

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

$\phi=12s$

$\phi =12(205)=2460$

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Example Question #681 : Rate Of Change

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's surface area to the rate of loss of its sides when its sides have length $\frac{7}{48}$ ?

Possible Answers:

$\frac{7}{8}$

$\frac{7}{192}$

$\frac{7}{96}$

$\frac{7}{4}$

$\frac{7}{2}$

Correct answer:

$\frac{7}{4}$

Explanation:

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

A=6s^2

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

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

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

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

$\phi=12s$

$\phi =12(\frac{7}{48})=\frac{7}{4}$

Report an Error

Example Question #681 : Rate Of Change

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's surface area to the rate of loss of its sides when its sides have length $\frac{11}{26}$ ?

Possible Answers:

$\frac{11}{13}$

$\frac{22}{13}$

$\frac{33}{13}$

$\frac{132}{13}$

$\frac{66}{13}$

Correct answer:

$\frac{66}{13}$

Explanation:

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

A=6s^2

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

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

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

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

$\phi=12s$

$\phi =12(\frac{11}{26})=\frac{66}{13}$

Report an Error

Example Question #683 : 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 surface area to the rate of loss of its sides when its sides have length $\frac{5}{33}$ ?

Possible Answers:

$\frac{40}{11}$

$\frac{20}{11}$

$\frac{10}{11}$

$\frac{20}{99}$

$\frac{20}{33}$

Correct answer:

$\frac{20}{11}$

Explanation:

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

A=6s^2

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

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

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

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

$\phi=12s$

$\phi =12(\frac{5}{33})=\frac{20}{11}$

Report an Error

Example Question #3601 : Calculus

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's surface area to the rate of loss of its sides when its sides have length $\frac{31}{21}$ ?

Possible Answers:

$\frac{124}{21}$

$\frac{124}{7}$

$\frac{248}{21}$

$\frac{31}{7}$

$\frac{93}{7}$

Correct answer:

$\frac{124}{7}$

Explanation:

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

A=6s^2

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

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

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

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

$\phi=12s$

$\phi =12(\frac{31}{21})=\frac{124}{7}$

Report an Error

Example Question #3602 : Calculus

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's surface area to the rate of loss of its sides when its sides have length $\frac{5}{39}$ ?

Possible Answers:

$\frac{10}{39}$

$\frac{10}{13}$

$\frac{20}{13}$

$\frac{20}{39}$

$\frac{40}{39}$

Correct answer:

$\frac{20}{13}$

Explanation:

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

A=6s^2

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

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

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

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

$\phi=12s$

$\phi =12(\frac{5}{39})=\frac{20}{13}$

Report an Error

Example Question #3603 : Calculus

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's surface area to the rate of loss of its sides when its sides have length $\frac{2}{87}$ ?

Possible Answers:

$\frac{2}{29}$

$\frac{16}{29}$

$\frac{8}{29}$

$\frac{1}{29}$

$\frac{4}{29}$

Correct answer:

$\frac{8}{29}$

Explanation:

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

A=6s^2

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

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

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

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

$\phi=12s$

$\phi =12(\frac{2}{87})=\frac{8}{29}$

Report an Error

Example Question #771 : Rate

A spherical balloon is being filled with air. What is ratio of the rate of growth of the volume of the sphere to the rate of growth of the surface area when the radius is 206?

Possible Answers:

103

824

1648

3296

412

Correct answer:

103

Explanation:

Let's begin by writing the equations for the volume and surface area of a sphere with respect to the sphere's radius:

$V=\frac{4}{3}\pi r^3$

$A=4\pi r^2$

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

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$

$\frac{dA}{dt}=8\pi r \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. To find the ratio of the rates of changes of the volume and surface area, divide:

$4\pi r^2 \frac{dr}{dt}=(\phi)8\pi r \frac{dr}{dt}$

$\phi =\frac{r}{2}$

$\phi =\frac{206}{2}=103$

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Calculus 1 : Rate

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #771 : Rate

Example Question #2561 : Functions

Example Question #2561 : Functions

Example Question #681 : Rate Of Change

Example Question #681 : Rate Of Change

Example Question #683 : How To Find Rate Of Change

Example Question #3601 : Calculus

Example Question #3602 : Calculus

Example Question #3603 : Calculus

Example Question #771 : Rate

All Calculus 1 Resources

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