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Calculus 1 : How to find rate of change

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Example Questions

Example Question #581 : Rate Of Change

A spherical balloon is being filled with air. What is the radius of the sphere at the instance the rate of growth of the surface area is 716 times the rate of growth of the circumference?

Possible Answers:

716

2864

179

1432

358

Correct answer:

179

Explanation:

Start by writing the equations for the surface area and circumference of a sphere with respect to the sphere's radius:

$A=4\pi r^2$

$C=2\pi r$

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

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

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. Now we can use the relation given in the problem statement, the rate of growth of the surface area is 716 times the rate of growth of the circumference, to solve for the length of the radius at that instant:

$8\pi r \frac{dr}{dt}=(316)2\pi \frac{dr}{dt}$

$r =\frac{716}{4}=179$

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Example Question #582 : Rate Of Change

A spherical balloon is being filled with air. What is the radius of the sphere at the instance the rate of growth of the surface area is 92 times the rate of growth of the circumference?

Possible Answers:

184

Correct answer:

Explanation:

Start by writing the equations for the surface area and circumference of a sphere with respect to the sphere's radius:

$A=4\pi r^2$

$C=2\pi r$

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

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

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. Now we can use the relation given in the problem statement, the rate of growth of the surface area is 92 times the rate of growth of the circumference, to solve for the length of the radius at that instant:

$8\pi r \frac{dr}{dt}=(92)2\pi \frac{dr}{dt}$

$r =\frac{92}{4}=23$

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Example Question #583 : Rate Of Change

A spherical balloon is being filled with air. What is the diameter of the sphere at the instance the rate of growth of the surface area is 1352 times the rate of growth of the circumference?

Possible Answers:

169

1690

338

676

845

Correct answer:

676

Explanation:

Start by writing the equations for the surface area and circumference of a sphere with respect to the sphere's radius:

$A=4\pi r^2$

$C=2\pi r$

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

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

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. Now we can use the relation given in the problem statement, the rate of growth of the surface area is 1352 times the rate of growth of the circumference, to solve for the length of the radius at that instant:

$8\pi r \frac{dr}{dt}=(1352)2\pi \frac{dr}{dt}$

$r =\frac{1352}{4}=338$

The diameter is then:

d=2r

d=676

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Example Question #584 : Rate Of Change

A spherical balloon is being filled with air. What is the diameter of the sphere at the instance the rate of growth of the surface area is 56 times the rate of growth of the circumference?

Possible Answers:

Correct answer:

Explanation:

Start by writing the equations for the surface area and circumference of a sphere with respect to the sphere's radius:

$A=4\pi r^2$

$C=2\pi r$

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

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

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. Now we can use the relation given in the problem statement, the rate of growth of the surface area is 56 times the rate of growth of the circumference, to solve for the length of the radius at that instant:

$8\pi r \frac{dr}{dt}=(56)2\pi \frac{dr}{dt}$

$r =\frac{56}{4}=14$

The diameter is then:

d=2r

d=28

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Example Question #585 : Rate Of Change

A spherical balloon is being filled with air. What is the diameter of the sphere at the instance the rate of growth of the surface area is 344 times the rate of growth of the circumference?

Possible Answers:

126

172

252

Correct answer:

172

Explanation:

Start by writing the equations for the surface area and circumference of a sphere with respect to the sphere's radius:

$A=4\pi r^2$

$C=2\pi r$

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

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

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. Now we can use the relation given in the problem statement, the rate of growth of the surface area is 344 times the rate of growth of the circumference, to solve for the length of the radius at that instant:

$8\pi r \frac{dr}{dt}=(344)2\pi \frac{dr}{dt}$

$r =\frac{344}{4}=86$

The diameter can then be found:

d=2r

d=172

Report an Error

Example Question #586 : Rate Of Change

A spherical balloon is being filled with air. What is the diameter of the sphere at the instance the rate of growth of the surface area is 1348 times the rate of growth of the circumference?

Possible Answers:

2696

185

337

674

370

Correct answer:

674

Explanation:

Start by writing the equations for the surface area and circumference of a sphere with respect to the sphere's radius:

$A=4\pi r^2$

$C=2\pi r$

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

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

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. Now we can use the relation given in the problem statement, the rate of growth of the surface area is 1348 times the rate of growth of the circumference, to solve for the length of the radius at that instant:

$8\pi r \frac{dr}{dt}=(1348)2\pi \frac{dr}{dt}$

$r =\frac{1348}{4}=337$

The diameter is then:

d=2r

d=674

Report an Error

Example Question #587 : Rate Of Change

A spherical balloon is being filled with air. What is the circumference of the sphere at the instance the rate of growth of the surface area is 232 times the rate of growth of the circumference?

Possible Answers:

$116\pi$

$464\pi$

$29\pi$

$58\pi$

$232\pi$

Correct answer:

$116\pi$

Explanation:

Start by writing the equations for the surface area and circumference of a sphere with respect to the sphere's radius:

$A=4\pi r^2$

$C=2\pi r$

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

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

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. Now we can use the relation given in the problem statement, the rate of growth of the surface area is 232 times the rate of growth of the circumference, to solve for the length of the radius at that instant:

$8\pi r \frac{dr}{dt}=(232)2\pi \frac{dr}{dt}$

$r =\frac{232}{4}=58$

The circumference is then:

$C=2\pi r$

$C=116\pi$

Report an Error

Example Question #671 : Rate

A spherical balloon is being filled with air. What is the circumference of the sphere at the instance the rate of growth of the surface area is 192 times the rate of growth of the circumference?

Possible Answers:

$768\pi$

$1536\pi$

$96\pi$

$384\pi$

$192\pi$

Correct answer:

$96\pi$

Explanation:

Start by writing the equations for the surface area and circumference of a sphere with respect to the sphere's radius:

$A=4\pi r^2$

$C=2\pi r$

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

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

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. Now we can use the relation given in the problem statement, the rate of growth of the surface area is 192 times the rate of growth of the circumference, to solve for the length of the radius at that instant:

$8\pi r \frac{dr}{dt}=(192)2\pi \frac{dr}{dt}$

$r =\frac{192}{4}=48$

The circumference can then be found at this instant:

$C=2\pi r$

$C=96\pi$

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Example Question #588 : Rate Of Change

A spherical balloon is being filled with air. What is the circumference of the sphere at the instance the rate of growth of the surface area is 1024 times the rate of growth of the circumference?

Possible Answers:

$4096\pi$

$1024\pi$

$128\pi$

$256\pi$

$512\pi$

Correct answer:

$512\pi$

Explanation:

Start by writing the equations for the surface area and circumference of a sphere with respect to the sphere's radius:

$A=4\pi r^2$

$C=2\pi r$

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

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

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. Now we can use the relation given in the problem statement, the rate of growth of the surface area is 1024 times the rate of growth of the circumference, to solve for the length of the radius at that instant:

$8\pi r \frac{dr}{dt}=(1024)2\pi \frac{dr}{dt}$

$r =\frac{1024}{4}=256$

The circumference is then:

$C=2\pi r$

$C=512\pi$

Report an Error

Example Question #589 : Rate Of Change

A spherical balloon is being filled with air. What is the circumference of the sphere at the instance the rate of growth of the surface area is 3132 times the rate of growth of the circumference?

Possible Answers:

$261\pi$

$522\pi$

$1566\pi$

$783\pi$

$1044\pi$

Correct answer:

$1566\pi$

Explanation:

Start by writing the equations for the surface area and circumference of a sphere with respect to the sphere's radius:

$A=4\pi r^2$

$C=2\pi r$

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

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

$\frac{dC}{dt}=2\pi \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere regardless of the considered parameter. Now we can use the relation given in the problem statement, the rate of growth of the surface area is 3132 times the rate of growth of the circumference, to solve for the length of the radius at that instant:

$8\pi r \frac{dr}{dt}=(3132)2\pi \frac{dr}{dt}$

$r =\frac{3132}{4}=783$

The circumference is then:

$C=2\pi r$

$C=1566\pi$

Report an Error

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Calculus 1 : How to find rate of change

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #581 : Rate Of Change

Example Question #582 : Rate Of Change

Example Question #583 : Rate Of Change

Example Question #584 : Rate Of Change

Example Question #585 : Rate Of Change

Example Question #586 : Rate Of Change

Example Question #587 : Rate Of Change

Example Question #671 : Rate

Example Question #588 : Rate Of Change

Example Question #589 : Rate Of Change

All Calculus 1 Resources

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