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 #751 : 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 diagonal when its sides have length $4\sqrt{33}$ ?

Possible Answers:

$48\sqrt{11}$

$132\sqrt{3}$

$44\sqrt{3}$

$44\sqrt{11}$

$48\sqrt{3}$

Correct answer:

$48\sqrt{11}$

Explanation:

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

A=6s^2

$d=s\sqrt{3}$

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

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

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

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

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(4\sqrt{33})\sqrt{3}=48\sqrt{11}$

Report an Error

Example Question #761 : 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 diagonal when its sides have length $11\sqrt{57}$ ?

Possible Answers:

$44\sqrt{19}$

$396\sqrt{57}$

$132\sqrt{19}$

$44\sqrt{57}$

$396\sqrt{19}$

Correct answer:

$132\sqrt{19}$

Explanation:

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

A=6s^2

$d=s\sqrt{3}$

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

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

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

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

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(11\sqrt{57})\sqrt{3}=132\sqrt{19}$

Report an Error

Example Question #762 : Rate

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 diagonal when its sides have length $\frac{\sqrt{3}}{54}$ ?

Possible Answers:

$\frac{1}{3}$

$\frac{8}{9}$

$\frac{4}{9}$

$\frac{4}{3}$

$\frac{2}{9}$

Correct answer:

$\frac{2}{9}$

Explanation:

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

A=6s^2

$d=s\sqrt{3}$

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

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

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

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

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(\frac{\sqrt{3}}{54})\sqrt{3}=\frac{2}{9}$

Report an Error

Example Question #763 : Rate

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 diagonal when its sides have length $\frac{\sqrt{6}}{16}$ ?

Possible Answers:

$2\sqrt{3}$

$\frac{3\sqrt{3}}{4}$

$\frac{3\sqrt{2}}{2}$

$4\sqrt{3}$

$\frac{3\sqrt{2}}{4}$

Correct answer:

$\frac{3\sqrt{2}}{4}$

Explanation:

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

A=6s^2

$d=s\sqrt{3}$

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

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

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

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

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(\frac{\sqrt{6}}{16})\sqrt{3}=\frac{3\sqrt{2}}{4}$

Report an Error

Example Question #671 : 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 diagonal when its sides have length $\frac{\sqrt{21}}{8}$ ?

Possible Answers:

$\frac{7\sqrt{3}}{8}$

$\frac{7\sqrt{3}}{4}$

$\frac{3\sqrt{7}}{2}$

$\frac{7\sqrt{3}}{2}$

$\frac{3\sqrt{7}}{4}$

Correct answer:

$\frac{3\sqrt{7}}{2}$

Explanation:

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

A=6s^2

$d=s\sqrt{3}$

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

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

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

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

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(\frac{\sqrt{21}}{8})\sqrt{3}=\frac{3\sqrt{7}}{2}$

Report an Error

Example Question #765 : Rate

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 diagonal when its sides have length $\frac{\sqrt{6}}{48}$ ?

Possible Answers:

$\frac{\sqrt{2}}{4}$

$\frac{3\sqrt{2}}{8}$

$\frac{\sqrt{2}}{2}$

$\frac{3\sqrt{2}}{4}$

$\frac{\sqrt{2}}{8}$

Correct answer:

$\frac{\sqrt{2}}{4}$

Explanation:

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

A=6s^2

$d=s\sqrt{3}$

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

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

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

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

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(\frac{\sqrt{6}}{48})\sqrt{3}=\frac{\sqrt{2}}{4}$

Report an Error

Example Question #671 : 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 diagonal when its sides have length $\frac{2}{19}$ ?

Possible Answers:

$\frac{6\sqrt{3}}{19}$

$\frac{16\sqrt{3}}{19}$

$\frac{4\sqrt{3}}{19}$

$\frac{2\sqrt{3}}{19}$

$\frac{8\sqrt{3}}{19}$

Correct answer:

$\frac{8\sqrt{3}}{19}$

Explanation:

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

A=6s^2

$d=s\sqrt{3}$

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

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

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

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

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(\frac{2}{19})\sqrt{3}=\frac{8\sqrt{3}}{19}$

Report an Error

Example Question #672 : 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 diagonal when its sides have length $\frac{\sqrt{7}}{7}$ ?

Possible Answers:

$\frac{4\sqrt{21}}{7}$

$\frac{2\sqrt{3}}{7}$

$4\sqrt{3}$

$\frac{4\sqrt{3}}{7}$

$2\sqrt{3}$

Correct answer:

$\frac{4\sqrt{21}}{7}$

Explanation:

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

A=6s^2

$d=s\sqrt{3}$

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

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

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

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

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(\frac{\sqrt{7}}{7})\sqrt{3}=\frac{4\sqrt{21}}{7}$

Report an Error

Example Question #673 : How To Find Rate Of Change

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 51?

Possible Answers:

816

612

306

1224

204

Correct answer:

612

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:

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

$\phi=12s$

$\phi =12(51)=612$

Report an Error

Example Question #674 : How To Find Rate Of Change

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 15?

Possible Answers:

240

180

480

120

360

Correct answer:

180

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:

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

$\phi=12s$

$\phi =12(15)=180$

Report an Error

View Calculus Tutors

Jethro
Certified Tutor

Federal University of Agriculture Abeokuta,, Bachelor of Engineering, Mechatronics, Robotics, and Automation Engineering.

View Calculus Tutors

Jozsef
Certified Tutor

University of Waterloo, Doctor of Philosophy, Applied Mathematics.

View Calculus Tutors

Laura
Certified Tutor

University of Cincinnati, Bachelor in Arts, Mathematics. University of Cincinnati, Master of Science, Mathematics.

All Calculus 1 Resources

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

Popular Subjects

Biology Tutors in Los Angeles, ISEE Tutors in Philadelphia, SSAT Tutors in Seattle, Physics Tutors in Dallas Fort Worth, Reading Tutors in Washington DC, GMAT Tutors in Washington DC, Computer Science Tutors in Atlanta, Statistics Tutors in Dallas Fort Worth, Biology Tutors in Denver, GRE Tutors in San Diego

Popular Courses & Classes

ACT Courses & Classes in Chicago, GMAT Courses & Classes in Miami, Spanish Courses & Classes in New York City, GMAT Courses & Classes in Atlanta, MCAT Courses & Classes in Seattle, SSAT Courses & Classes in Denver, MCAT Courses & Classes in Denver, ACT Courses & Classes in Phoenix, GRE Courses & Classes in Phoenix, Spanish Courses & Classes in Atlanta

Popular Test Prep

MCAT Test Prep in Denver, GRE Test Prep in Houston, LSAT Test Prep in San Francisco-Bay Area, ISEE Test Prep in Atlanta, ISEE Test Prep in Denver, SAT Test Prep in Miami, LSAT Test Prep in Seattle, ACT Test Prep in Chicago, GMAT Test Prep in Seattle, ACT Test Prep in Phoenix

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 #751 : Rate

Example Question #761 : Rate

Example Question #762 : Rate

Example Question #763 : Rate

Example Question #671 : Rate Of Change

Example Question #765 : Rate

Example Question #671 : How To Find Rate Of Change

Example Question #672 : How To Find Rate Of Change

Example Question #673 : How To Find Rate Of Change

Example Question #674 : How To Find 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: