We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Rate of Change

Study concepts, example questions & explanations for Calculus 1

Create An Account

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #3334 : Calculus

A spherical balloon is deflating, while maintaining its spherical shape. What is the diameter of the sphere at the instance the rate of shrinkage of the surface area is 0.08 times the rate of shrinkage of the circumference?

Possible Answers:

0.04

0.01

0.02

0.16

0.08

Correct answer:

0.04

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 shrinkage of the surface area is 0.08 times the rate of shrinkage of the circumference, to solve for the length of the radius at that instant:

$8\pi r \frac{dr}{dt}=(0.08)2\pi \frac{dr}{dt}$

$r =\frac{0.08}{4}=0.02$

The diameter is then:

d=2r

d=0.04

Report an Error

Example Question #421 : How To Find Rate Of Change

Find the equation for the rate of change of the volume of a cylindrical pitcher with respect to radius. The height of the cylinder is three times the size of the radius.

Possible Answers:

$\frac{dV}{dr}=9\pi r^{2}$

$\frac{dV}{dr}=9\pi r^{2}h$

$\frac{dV}{dr}=3\pi r^{2}$

$\frac{dV}{dr}=3\pi r^{2}h$

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

Correct answer:

$\frac{dV}{dr}=9\pi r^{2}$

Explanation:

The volume of a cylinder is $\pi r^{2}h$ . Before we take the derivative of the volume equation, we must first make all of our variable in terms of radius. Therefore, $V=\pi r^{2}(3r)=3\pi r^{3}$ .

Now, we take the derivative with respect to radius: $dV=9 \pi r^{2}dr.$

Rearrange to get: $\frac{dV}{dr}=9 \pi r^{2}.$

Report an Error

Example Question #3335 : 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 92?

Possible Answers:

184

61.3

Correct answer:

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:

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

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

$\phi =\frac{92}{2}=46$

Report an Error

Example Question #3336 : 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 140?

Possible Answers:

140

Correct answer:

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:

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

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

$\phi =\frac{140}{2}=70$

Report an Error

Example Question #424 : 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 162?

Possible Answers:

216

108

Correct answer:

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:

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

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

$\phi =\frac{162}{2}=81$

Report an Error

Example Question #425 : How To Find Rate Of Change

A spherical balloon is deflating, although it maintains its spherical shape. What is ratio of the rate of loss of the volume of the sphere to the rate of loss of the surface area when the radius is 910?

Possible Answers:

130

455

520

260

Correct answer:

455

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:

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

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

$\phi =\frac{910}{2}=455$

Report an Error

Example Question #3337 : Calculus

Possible Answers:

26.5

79.5

106

318

Correct answer:

79.5

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:

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

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

$\phi =\frac{159}{2}=79.5$

Report an Error

Example Question #421 : Rate Of Change

Possible Answers:

310

248

124

496

Correct answer:

496

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:

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

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

$\phi =\frac{992}{2}=496$

Report an Error

Example Question #422 : 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 radius when the radius is 1?

Possible Answers:

$4\pi$

$4\pi^2$

$2\pi$

$2\pi^2$

Correct answer:

$4\pi$

Explanation:

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

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

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

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

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 radius, divide:

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

$\phi =4\pi r^2$

$\phi =4\pi(1)^2=4\pi$

Report an Error

Example Question #423 : 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 radius when the radius is 0.25?

Possible Answers:

$\pi$

$\frac{\pi}{8}$

$\frac{\pi}{2}$

$\frac{\pi}{4}$

$\frac{\pi}{16}$

Correct answer:

$\frac{\pi}{4}$

Explanation:

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

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

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

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

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 radius, divide:

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

$\phi =4\pi r^2$

$\phi =4\pi(0.25)^2=\frac{\pi}{4}$

Report an Error

View Calculus Tutors

Leonard
Certified Tutor

Florida State University, Bachelor of Science, Environmental Science.

View Calculus Tutors

Soung
Certified Tutor

University of Chicago, Bachelor in Arts, Chemistry. Northwestern University, Doctor of Science, Molecular Pharmacology.

View Calculus Tutors

Audrey
Certified Tutor

SUNY at Binghamton, Bachelor in Arts, Actuarial Science.

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Spanish Tutors in Atlanta, MCAT Tutors in San Francisco-Bay Area, GMAT Tutors in Los Angeles, Algebra Tutors in San Francisco-Bay Area, LSAT Tutors in Boston, Biology Tutors in New York City, GRE Tutors in New York City, SAT Tutors in Los Angeles, Spanish Tutors in Dallas Fort Worth, Math Tutors in Miami

Popular Courses & Classes

ACT Courses & Classes in San Diego, MCAT Courses & Classes in Chicago, GMAT Courses & Classes in Atlanta, SSAT Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in New York City, LSAT Courses & Classes in Boston, SSAT Courses & Classes in Atlanta, GMAT Courses & Classes in Washington DC, Spanish Courses & Classes in Seattle, SSAT Courses & Classes in San Diego

Popular Test Prep

SSAT Test Prep in Chicago, GMAT Test Prep in Atlanta, ACT Test Prep in Washington DC, MCAT Test Prep in Los Angeles, MCAT Test Prep in San Diego, LSAT Test Prep in Phoenix, GMAT Test Prep in Chicago, SSAT Test Prep in Boston, ISEE Test Prep in San Francisco-Bay Area, MCAT Test Prep in Boston

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Calculus 1 : Rate of Change

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #3334 : Calculus

Example Question #421 : How To Find Rate Of Change

Example Question #3335 : Calculus

Example Question #3336 : Calculus

Example Question #424 : How To Find Rate Of Change

Example Question #425 : How To Find Rate Of Change

Example Question #3337 : Calculus

Example Question #421 : Rate Of Change

Example Question #422 : Rate Of Change

Example Question #423 : Rate Of Change

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: