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 #661 : 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 volume to the rate of growth of its sides when its sides have length 30?

Possible Answers:

900 $\displaystyle 900$

2700 $\displaystyle 2700$

5400 $\displaystyle 5400$

8100 $\displaystyle 8100$

$\displaystyle 90$

Correct answer:

2700 $\displaystyle 2700$

Explanation:

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

V=s^3 $\displaystyle V=s^3$

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$ $\displaystyle \frac{dV}{dt}=3s^2\frac{ds}{dt}$

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

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$ $\displaystyle 3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$ $\displaystyle \phi=3s^2$

$\phi =3(30)^2=2700$ $\displaystyle \phi =3(30)^2=2700$

Report an Error

Example Question #662 : 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 volume to the rate of growth of its sides when its sides have length 32?

Possible Answers:

3072 $\displaystyle 3072$

9216 $\displaystyle 9216$

1024 $\displaystyle 1024$

$\displaystyle 96$

27648 $\displaystyle 27648$

Correct answer:

3072 $\displaystyle 3072$

Explanation:

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

V=s^3 $\displaystyle V=s^3$

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$ $\displaystyle \frac{dV}{dt}=3s^2\frac{ds}{dt}$

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

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$ $\displaystyle 3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$ $\displaystyle \phi=3s^2$

$\phi =3(32)^2=3072$ $\displaystyle \phi =3(32)^2=3072$

Report an Error

Example Question #663 : 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 volume to the rate of growth of its sides when its sides have length 95?

Possible Answers:

285 $\displaystyle 285$

855 $\displaystyle 855$

27075 $\displaystyle 27075$

9025 $\displaystyle 9025$

81225 $\displaystyle 81225$

Correct answer:

27075 $\displaystyle 27075$

Explanation:

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

V=s^3 $\displaystyle V=s^3$

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$ $\displaystyle \frac{dV}{dt}=3s^2\frac{ds}{dt}$

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

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$ $\displaystyle 3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$ $\displaystyle \phi=3s^2$

$\phi =3(95)^2=27075$ $\displaystyle \phi =3(95)^2=27075$

Report an Error

Example Question #664 : 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 volume to the rate of loss of its sides when its sides have length $\frac{14}{15}$ $\displaystyle \frac{14}{15}$?

Possible Answers:

$\frac{14}{225}$ $\displaystyle \frac{14}{225}$

$\frac{196}{225}$ $\displaystyle \frac{196}{225}$

$\frac{14}{75}$ $\displaystyle \frac{14}{75}$

$\frac{196}{25}$ $\displaystyle \frac{196}{25}$

$\frac{196}{75}$ $\displaystyle \frac{196}{75}$

Correct answer:

$\frac{196}{75}$ $\displaystyle \frac{196}{75}$

Explanation:

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

V=s^3 $\displaystyle V=s^3$

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$ $\displaystyle \frac{dV}{dt}=3s^2\frac{ds}{dt}$

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

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$ $\displaystyle 3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$ $\displaystyle \phi=3s^2$

$\phi =3(\frac{14}{15})^2=\frac{196}{75}$ $\displaystyle \phi =3(\frac{14}{15})^2=\frac{196}{75}$

Report an Error

Example Question #665 : 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 volume to the rate of loss of its sides when its sides have length $\frac{13}{9}$ $\displaystyle \frac{13}{9}$?

Possible Answers:

$\frac{169}{9}$ $\displaystyle \frac{169}{9}$

$\frac{169}{27}$ $\displaystyle \frac{169}{27}$

$\frac{13}{81}$ $\displaystyle \frac{13}{81}$

$\frac{169}{81}$ $\displaystyle \frac{169}{81}$

$\frac{13}{27}$ $\displaystyle \frac{13}{27}$

Correct answer:

$\frac{169}{27}$ $\displaystyle \frac{169}{27}$

Explanation:

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

V=s^3 $\displaystyle V=s^3$

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$ $\displaystyle \frac{dV}{dt}=3s^2\frac{ds}{dt}$

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

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$ $\displaystyle 3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$ $\displaystyle \phi=3s^2$

$\phi =3(\frac{13}{9})^2=\frac{169}{27}$ $\displaystyle \phi =3(\frac{13}{9})^2=\frac{169}{27}$

Report an Error

Example Question #666 : 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 volume to the rate of loss of its sides when its sides have length $\frac{11}{4}$ $\displaystyle \frac{11}{4}$?

Possible Answers:

$\frac{121}{8}$ $\displaystyle \frac{121}{8}$

$\frac{363}{16}$ $\displaystyle \frac{363}{16}$

$\frac{121}{4}$ $\displaystyle \frac{121}{4}$

$\frac{121}{16}$ $\displaystyle \frac{121}{16}$

$\frac{363}{4}$ $\displaystyle \frac{363}{4}$

Correct answer:

$\frac{363}{16}$ $\displaystyle \frac{363}{16}$

Explanation:

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

V=s^3 $\displaystyle V=s^3$

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$ $\displaystyle \frac{dV}{dt}=3s^2\frac{ds}{dt}$

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

$3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$ $\displaystyle 3s^2\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=3s^2$ $\displaystyle \phi=3s^2$

$\phi =3(\frac{11}{4})^2=\frac{363}{16}$ $\displaystyle \phi =3(\frac{11}{4})^2=\frac{363}{16}$

Report an Error

Example Question #667 : 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 diagonal when its sides have length $11\sqrt{17}$ $\displaystyle 11\sqrt{17}$?

Possible Answers:

$132\sqrt{3}$ $\displaystyle 132\sqrt{3}$

$17\sqrt{51}$ $\displaystyle 17\sqrt{51}$

$44\sqrt{51}$ $\displaystyle 44\sqrt{51}$

$33\sqrt{51}$ $\displaystyle 33\sqrt{51}$

$132\sqrt{17}$ $\displaystyle 132\sqrt{17}$

Correct answer:

$44\sqrt{51}$ $\displaystyle 44\sqrt{51}$

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 $\displaystyle A=6s^2$

$d=s\sqrt{3}$ $\displaystyle d=s\sqrt{3}$

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

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

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

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

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

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

$\phi =4(11\sqrt{17})\sqrt{3}=44\sqrt{51}$ $\displaystyle \phi =4(11\sqrt{17})\sqrt{3}=44\sqrt{51}$

Report an Error

Example Question #668 : 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 diagonal when its sides have length $2\sqrt{10}$ $\displaystyle 2\sqrt{10}$?

Possible Answers:

$12\sqrt{15}$ $\displaystyle 12\sqrt{15}$

$4\sqrt{15}$ $\displaystyle 4\sqrt{15}$

$80\sqrt{3}$ $\displaystyle 80\sqrt{3}$

$8\sqrt{30}$ $\displaystyle 8\sqrt{30}$

$16\sqrt{15}$ $\displaystyle 16\sqrt{15}$

Correct answer:

$8\sqrt{30}$ $\displaystyle 8\sqrt{30}$

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 $\displaystyle A=6s^2$

$d=s\sqrt{3}$ $\displaystyle d=s\sqrt{3}$

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

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

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

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

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

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

$\phi =4(2\sqrt{10})\sqrt{3}=8\sqrt{30}$ $\displaystyle \phi =4(2\sqrt{10})\sqrt{3}=8\sqrt{30}$

Report an Error

Example Question #669 : 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 diagonal when its sides have length $2\sqrt{42}$ $\displaystyle 2\sqrt{42}$?

Possible Answers:

$7\sqrt{51}$ $\displaystyle 7\sqrt{51}$

$24\sqrt{14}$ $\displaystyle 24\sqrt{14}$

$16\sqrt{42}$ $\displaystyle 16\sqrt{42}$

$8\sqrt{42}$ $\displaystyle 8\sqrt{42}$

$12\sqrt{14}$ $\displaystyle 12\sqrt{14}$

Correct answer:

$24\sqrt{14}$ $\displaystyle 24\sqrt{14}$

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 $\displaystyle A=6s^2$

$d=s\sqrt{3}$ $\displaystyle d=s\sqrt{3}$

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

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

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

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

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

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

$\phi =4(2\sqrt{42})\sqrt{3}=24\sqrt{14}$ $\displaystyle \phi =4(2\sqrt{42})\sqrt{3}=24\sqrt{14}$

Report an Error

Example Question #670 : 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 diagonal when its sides have length $3\sqrt{39}$ $\displaystyle 3\sqrt{39}$?

Possible Answers:

$108\sqrt{13}$ $\displaystyle 108\sqrt{13}$

$12\sqrt{13}$ $\displaystyle 12\sqrt{13}$

$36\sqrt{39}$ $\displaystyle 36\sqrt{39}$

$36\sqrt{13}$ $\displaystyle 36\sqrt{13}$

$12\sqrt{39}$ $\displaystyle 12\sqrt{39}$

Correct answer:

$36\sqrt{13}$ $\displaystyle 36\sqrt{13}$

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 $\displaystyle A=6s^2$

$d=s\sqrt{3}$ $\displaystyle d=s\sqrt{3}$

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

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

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

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

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

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

$\phi =4(3\sqrt{39})\sqrt{3}=36\sqrt{13}$ $\displaystyle \phi =4(3\sqrt{39})\sqrt{3}=36\sqrt{13}$

Report an Error

View Calculus Tutors

Derick
Certified Tutor

Northeastern University, Bachelor of Engineering, Mechanical Engineering.

View Calculus Tutors

Robert
Certified Tutor

University of Guelph, Bachelor, Biochemistry.

View Calculus Tutors

Jerry
Certified Tutor

University of Illinois at Chicago, Bachelor of Science, Engineering Physics. University of Illinois at Chicago, Master of Sci...

All Calculus 1 Resources

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

Popular Subjects

Statistics Tutors in Washington DC, GRE Tutors in Philadelphia, Computer Science Tutors in Phoenix, LSAT Tutors in Washington DC, Math Tutors in San Francisco-Bay Area, French Tutors in Boston, Spanish Tutors in Denver, Biology Tutors in Boston, ACT Tutors in Washington DC, English Tutors in San Francisco-Bay Area

Popular Courses & Classes

Spanish Courses & Classes in Seattle, ISEE Courses & Classes in Miami, SAT Courses & Classes in Phoenix, SSAT Courses & Classes in Atlanta, ISEE Courses & Classes in Boston, GRE Courses & Classes in Houston, LSAT Courses & Classes in San Diego, SSAT Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Miami, LSAT Courses & Classes in Atlanta

Popular Test Prep

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

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 #661 : How To Find Rate Of Change

Example Question #662 : How To Find Rate Of Change

Example Question #663 : How To Find Rate Of Change

Example Question #664 : How To Find Rate Of Change

Example Question #665 : How To Find Rate Of Change

Example Question #666 : How To Find Rate Of Change

Example Question #667 : How To Find Rate Of Change

Example Question #668 : How To Find Rate Of Change

Example Question #669 : How To Find Rate Of Change

Example Question #670 : 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: