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Calculus 1 : Functions

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Example Questions

Example Question #2571 : Functions

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 $\frac{5}{33}$ $\displaystyle \frac{5}{33}$?

Possible Answers:

$\frac{20}{11}$ $\displaystyle \frac{20}{11}$

$\frac{40}{11}$ $\displaystyle \frac{40}{11}$

$\frac{20}{33}$ $\displaystyle \frac{20}{33}$

$\frac{20}{99}$ $\displaystyle \frac{20}{99}$

$\frac{10}{11}$ $\displaystyle \frac{10}{11}$

Correct answer:

$\frac{20}{11}$ $\displaystyle \frac{20}{11}$

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 $\displaystyle A=6s^2$

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

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

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

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

$\phi=12s$ $\displaystyle \phi=12s$

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

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Example Question #2572 : Functions

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 $\frac{31}{21}$ $\displaystyle \frac{31}{21}$?

Possible Answers:

$\frac{93}{7}$ $\displaystyle \frac{93}{7}$

$\frac{124}{21}$ $\displaystyle \frac{124}{21}$

$\frac{124}{7}$ $\displaystyle \frac{124}{7}$

$\frac{31}{7}$ $\displaystyle \frac{31}{7}$

$\frac{248}{21}$ $\displaystyle \frac{248}{21}$

Correct answer:

$\frac{124}{7}$ $\displaystyle \frac{124}{7}$

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 $\displaystyle A=6s^2$

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

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

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

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

$\phi=12s$ $\displaystyle \phi=12s$

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

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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 $\frac{5}{39}$ $\displaystyle \frac{5}{39}$?

Possible Answers:

$\frac{10}{39}$ $\displaystyle \frac{10}{39}$

$\frac{20}{13}$ $\displaystyle \frac{20}{13}$

$\frac{20}{39}$ $\displaystyle \frac{20}{39}$

$\frac{40}{39}$ $\displaystyle \frac{40}{39}$

$\frac{10}{13}$ $\displaystyle \frac{10}{13}$

Correct answer:

$\frac{20}{13}$ $\displaystyle \frac{20}{13}$

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 $\displaystyle A=6s^2$

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

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

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

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

$\phi=12s$ $\displaystyle \phi=12s$

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

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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 $\frac{2}{87}$ $\displaystyle \frac{2}{87}$?

Possible Answers:

$\frac{2}{29}$ $\displaystyle \frac{2}{29}$

$\frac{16}{29}$ $\displaystyle \frac{16}{29}$

$\frac{8}{29}$ $\displaystyle \frac{8}{29}$

$\frac{1}{29}$ $\displaystyle \frac{1}{29}$

$\frac{4}{29}$ $\displaystyle \frac{4}{29}$

Correct answer:

$\frac{8}{29}$ $\displaystyle \frac{8}{29}$

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 $\displaystyle A=6s^2$

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

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

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

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

$\phi=12s$ $\displaystyle \phi=12s$

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

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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 $\displaystyle 103$

824 $\displaystyle 824$

1648 $\displaystyle 1648$

3296 $\displaystyle 3296$

412 $\displaystyle 412$

Correct answer:

103 $\displaystyle 103$

Explanation:

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

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

$A=4\pi r^2$ $\displaystyle A=4\pi r^2$

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

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

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

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:

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

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

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

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Example Question #3605 : Calculus

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 276?

Possible Answers:

138 $\displaystyle 138$

$\displaystyle 69$

$\displaystyle 92$

$\displaystyle 46$

184 $\displaystyle 184$

Correct answer:

138 $\displaystyle 138$

Explanation:

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

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

$A=4\pi r^2$ $\displaystyle A=4\pi r^2$

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

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

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

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:

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

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

$\phi =\frac{276}{2}=138$ $\displaystyle \phi =\frac{276}{2}=138$

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Example Question #691 : How To Find Rate Of Change

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 2112?

Possible Answers:

1056 $\displaystyle 1056$

176 $\displaystyle 176$

704 $\displaystyle 704$

264 $\displaystyle 264$

528 $\displaystyle 528$

Correct answer:

1056 $\displaystyle 1056$

Explanation:

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

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

$A=4\pi r^2$ $\displaystyle A=4\pi r^2$

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

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

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

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:

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

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

$\phi =\frac{2112}{2}=1056$ $\displaystyle \phi =\frac{2112}{2}=1056$

Report an Error

Example Question #691 : Rate Of Change

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 2376?

Possible Answers:

396 $\displaystyle 396$

1188 $\displaystyle 1188$

108 $\displaystyle 108$

594 $\displaystyle 594$

792 $\displaystyle 792$

Correct answer:

1188 $\displaystyle 1188$

Explanation:

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

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

$A=4\pi r^2$ $\displaystyle A=4\pi r^2$

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

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

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

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:

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

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

$\phi =\frac{2376}{2}=1188$ $\displaystyle \phi =\frac{2376}{2}=1188$

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Example Question #2573 : Functions

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 3112?

Possible Answers:

12448 $\displaystyle 12448$

1556 $\displaystyle 1556$

4149 $\displaystyle 4149$

778 $\displaystyle 778$

6224 $\displaystyle 6224$

Correct answer:

1556 $\displaystyle 1556$

Explanation:

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

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

$A=4\pi r^2$ $\displaystyle A=4\pi r^2$

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

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

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

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:

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

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

$\phi =\frac{3112}{2}=1556$ $\displaystyle \phi =\frac{3112}{2}=1556$

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Example Question #3609 : Calculus

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 1480?

Possible Answers:

740 $\displaystyle 740$

2220 $\displaystyle 2220$

370 $\displaystyle 370$

2960 $\displaystyle 2960$

1110 $\displaystyle 1110$

Correct answer:

740 $\displaystyle 740$

Explanation:

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

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

$A=4\pi r^2$ $\displaystyle A=4\pi r^2$

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

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

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

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:

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

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

$\phi =\frac{1480}{2}=740$ $\displaystyle \phi =\frac{1480}{2}=740$

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Calculus 1 : Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #2571 : Functions

Example Question #2572 : Functions

Example Question #681 : Rate Of Change

Example Question #3603 : Calculus

Example Question #771 : Rate

Example Question #3605 : Calculus

Example Question #691 : How To Find Rate Of Change

Example Question #691 : Rate Of Change

Example Question #2573 : Functions

Example Question #3609 : Calculus

All Calculus 1 Resources

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