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Calculus 1 : Rate

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Example Questions

Example Question #841 : Rate

A spherical balloon is being filled with air. What is the rate of growth of the sphere's surface area when the radius is 3 and the rate of change of the radius is 15?

Possible Answers:

$90\pi$ $\displaystyle 90\pi$

$45\pi$ $\displaystyle 45\pi$

$3240\pi$ $\displaystyle 3240\pi$

$360\pi$ $\displaystyle 360\pi$

$1080\pi$ $\displaystyle 1080\pi$

Correct answer:

$360\pi$ $\displaystyle 360\pi$

Explanation:

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

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

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

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

Considering what was given as our problem conditions, the radius is 3 and the rate of change of the radius is 15, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(3)(15)=360\pi$ $\displaystyle \frac{dA}{dt}=8\pi(3)(15)=360\pi$

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Example Question #751 : Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's surface area when the radius is 20 and the rate of change of the radius is 1?

Possible Answers:

$3200\pi$ $\displaystyle 3200\pi$

$80\pi$ $\displaystyle 80\pi$

$50\pi$ $\displaystyle 50\pi$

$400\pi$ $\displaystyle 400\pi$

$160\pi$ $\displaystyle 160\pi$

Correct answer:

$160\pi$ $\displaystyle 160\pi$

Explanation:

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

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

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

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

Considering what was given as our problem conditions, the radius is 20 and the rate of change of the radius is 1, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(20)(1)=160\pi$ $\displaystyle \frac{dA}{dt}=8\pi(20)(1)=160\pi$

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Example Question #841 : Rate

A spherical balloon is being filled with air. What is the rate of growth of the sphere's surface area when the radius is 23 and the rate of change of the radius is 4?

Possible Answers:

$1296\pi$ $\displaystyle 1296\pi$

$16928\pi$ $\displaystyle 16928\pi$

$2116\pi$ $\displaystyle 2116\pi$

$92\pi$ $\displaystyle 92\pi$

$736\pi$ $\displaystyle 736\pi$

Correct answer:

$736\pi$ $\displaystyle 736\pi$

Explanation:

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

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

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

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

Considering what was given as our problem conditions, the radius is 23 and the rate of change of the radius is 4, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(23)(4)=736\pi$ $\displaystyle \frac{dA}{dt}=8\pi(23)(4)=736\pi$

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Example Question #3661 : Calculus

A spherical balloon is being filled with air. What is the rate of growth of the sphere's surface area when the radius is 2 and the rate of change of the radius is 21?

Possible Answers:

$14112\pi$ $\displaystyle 14112\pi$

$42\pi$ $\displaystyle 42\pi$

$1764\pi$ $\displaystyle 1764\pi$

$672\pi$ $\displaystyle 672\pi$

$336\pi$ $\displaystyle 336\pi$

Correct answer:

$336\pi$ $\displaystyle 336\pi$

Explanation:

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

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

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

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

Considering what was given as our problem conditions, the radius is 2 and the rate of change of the radius is 21, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(2)(21)=336\pi$ $\displaystyle \frac{dA}{dt}=8\pi(2)(21)=336\pi$

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Example Question #754 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's surface area when the radius is 4 and the rate of change of the radius is 21?

Possible Answers:

$84\pi$ $\displaystyle 84\pi$

$2688\pi$ $\displaystyle 2688\pi$

$42\pi$ $\displaystyle 42\pi$

$336\pi$ $\displaystyle 336\pi$

$672\pi$ $\displaystyle 672\pi$

Correct answer:

$672\pi$ $\displaystyle 672\pi$

Explanation:

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

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

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

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

Considering what was given as our problem conditions, the radius is 4 and the rate of change of the radius is 21, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(4)(21)=672\pi$ $\displaystyle \frac{dA}{dt}=8\pi(4)(21)=672\pi$

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Example Question #755 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's surface area when the radius is 3 and the rate of change of the radius is 23?

Possible Answers:

$12696\pi$ $\displaystyle 12696\pi$

$552\pi$ $\displaystyle 552\pi$

$38088\pi$ $\displaystyle 38088\pi$

$69\pi$ $\displaystyle 69\pi$

$4761\pi$ $\displaystyle 4761\pi$

Correct answer:

$552\pi$ $\displaystyle 552\pi$

Explanation:

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

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

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

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

Considering what was given as our problem conditions, the radius is 3 and the rate of change of the radius is 23, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(3)(23)=552\pi$ $\displaystyle \frac{dA}{dt}=8\pi(3)(23)=552\pi$

Report an Error

Example Question #756 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's surface area when the radius is 2 and the rate of change of the radius is 23?

Possible Answers:

$736\pi$ $\displaystyle 736\pi$

$368\pi$ $\displaystyle 368\pi$

$46\pi$ $\displaystyle 46\pi$

$16928\pi$ $\displaystyle 16928\pi$

$8464\pi$ $\displaystyle 8464\pi$

Correct answer:

$368\pi$ $\displaystyle 368\pi$

Explanation:

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

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

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

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

Considering what was given as our problem conditions, the radius is 2 and the rate of change of the radius is 23, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(2)(23)=368\pi$ $\displaystyle \frac{dA}{dt}=8\pi(2)(23)=368\pi$

Report an Error

Example Question #751 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's surface area when the radius is 21 and the rate of change of the radius is 2?

Possible Answers:

$42\pi$ $\displaystyle 42\pi$

$168\pi$ $\displaystyle 168\pi$

$336\pi$ $\displaystyle 336\pi$

$21\pi$ $\displaystyle 21\pi$

$84\pi$ $\displaystyle 84\pi$

Correct answer:

$336\pi$ $\displaystyle 336\pi$

Explanation:

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

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

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

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

Considering what was given as our problem conditions, the radius is 21 and the rate of change of the radius is 2, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(21)(2)=336\pi$ $\displaystyle \frac{dA}{dt}=8\pi(21)(2)=336\pi$

Report an Error

Example Question #758 : How To Find Rate Of Change

A spherical balloon is being filled with air. What is the rate of growth of the sphere's surface area when the radius is 4 and the rate of change of the radius is 10?

Possible Answers:

$40\pi$ $\displaystyle 40\pi$

$1280\pi$ $\displaystyle 1280\pi$

$1600\pi$ $\displaystyle 1600\pi$

$320\pi$ $\displaystyle 320\pi$

$3200\pi$ $\displaystyle 3200\pi$

Correct answer:

$320\pi$ $\displaystyle 320\pi$

Explanation:

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

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

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

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

Considering what was given as our problem conditions, the radius is 4 and the rate of change of the radius is 10, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(4)10()=320\pi$ $\displaystyle \frac{dA}{dt}=8\pi(4)10()=320\pi$

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Example Question #841 : Rate

A spherical balloon is being filled with air. What is the rate of growth of the sphere's surface area when the radius is 2 and the rate of change of the radius is 14?

Possible Answers:

$3136\pi$ $\displaystyle 3136\pi$

$392\pi$ $\displaystyle 392\pi$

$224\pi$ $\displaystyle 224\pi$

$56\pi$ $\displaystyle 56\pi$

$28\pi$ $\displaystyle 28\pi$

Correct answer:

$224\pi$ $\displaystyle 224\pi$

Explanation:

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

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

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

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

Considering what was given as our problem conditions, the radius is 2 and the rate of change of the radius is 14, we can now find the rate of change of the surface area:

$\frac{dA}{dt}=8\pi(2)(14)=224\pi$ $\displaystyle \frac{dA}{dt}=8\pi(2)(14)=224\pi$

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Calculus 1 : Rate

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #841 : Rate

Example Question #751 : Rate Of Change

Example Question #841 : Rate

Example Question #3661 : Calculus

Example Question #754 : How To Find Rate Of Change

Example Question #755 : How To Find Rate Of Change

Example Question #756 : How To Find Rate Of Change

Example Question #751 : How To Find Rate Of Change

Example Question #758 : How To Find Rate Of Change

Example Question #841 : Rate

All Calculus 1 Resources

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