We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Calculus

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 #562 : Rate

A cube is growing in size. What is the length of the sides of the cube at the time that the rate of growth of the cube's volume is equal to 30 times the rate of growth of its surface area?

Possible Answers:

360

540

720

120

Correct answer:

120

Explanation:

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

V=s^3

A=6s^2

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

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

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

Now, solve for the length of the side of the cube to satisfy the problem condition, the rate of growth of the cube's volume is equal to 30 times the rate of growth of its surface area:

$3s^2\frac{ds}{dt}=(30)12s\frac{ds}{dt}$

s=(30)4=120

Report an Error

Example Question #563 : Rate

A cube is growing in size. What is the volume of the cube at the time that the rate of growth of the cube's volume is equal to $\frac{1}{8}$ times the rate of growth of its surface area?

Possible Answers:

$\frac{1}{16}$

$\frac{1}{8}$

$\frac{1}{32}$

$\frac{1}{4}$

$\frac{1}{2}$

Correct answer:

$\frac{1}{8}$

Explanation:

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

V=s^3

A=6s^2

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

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

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

Now, solve for the length of the side of the cube to satisfy the problem condition, the rate of growth of the cube's volume is equal to $\frac{1}{8}$ times the rate of growth of its surface area:

$3s^2\frac{ds}{dt}=(\frac{1}{8})12s\frac{ds}{dt}$

$s=(\frac{1}{8})4=\frac{1}{2}$

Now to find the volume

V=s^3

$V=(\frac{1}{2})^3=\frac{1}{8}$

Report an Error

Example Question #564 : Rate

A cube is growing in size. What is the surface area of the cube at the time that the rate of growth of the cube's volume is equal to $\frac{2}{3}$ times the rate of growth of its surface area?

Possible Answers:

$\frac{16}{3}$

$\frac{32}{3}$

$\frac{64}{3}$

$\frac{8}{3}$

$\frac{128}{3}$

Correct answer:

$\frac{128}{3}$

Explanation:

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

V=s^3

A=6s^2

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

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

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

Now, solve for the length of the side of the cube to satisfy the problem condition, the rate of growth of the cube's volume is equal to $\frac{2}{3}$ times the rate of growth of its surface area:

$3s^2\frac{ds}{dt}=(\frac{2}{3})12s\frac{ds}{dt}$

$s=(\frac{2}{3})4=\frac{8}{3}$

Then to find the surface area:

A=6s^2

$A=6(\frac{8}{3})^2=\frac{128}{3}$

Report an Error

Example Question #481 : How To Find Rate Of Change

A cube is growing in size. What is the surface area of the cube at the time that the rate of growth of the cube's volume is equal to $\frac{1}{96}$ times the rate of growth of its surface area?

Possible Answers:

$\frac{1}{16}$

$\frac{1}{72}$

$\frac{1}{24}$

$\frac{1}{96}$

$\frac{1}{256}$

Correct answer:

$\frac{1}{96}$

Explanation:

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

V=s^3

A=6s^2

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

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

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

Now, solve for the length of the side of the cube to satisfy the problem condition, the rate of growth of the cube's volume is equal to $\frac{1}{96}$ times the rate of growth of its surface area:

$3s^2\frac{ds}{dt}=(\frac{1}{96})12s\frac{ds}{dt}$

$s=(\frac{1}{96})4=\frac{1}{24}$

Then to find the surface area:

A=6s^2

$A=6(\frac{1}{24})^2=\frac{1}{96}$

Report an Error

Example Question #482 : How To Find Rate Of Change

A cube is growing in size. What is the area of a face of the cube at the time that the rate of growth of the cube's volume is equal to $\frac{1}{16}$ times the rate of growth of its surface area?

Possible Answers:

$\frac{1}{16}$

$\frac{1}{4}$

$\frac{1}{8}$

Correct answer:

$\frac{1}{16}$

Explanation:

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

V=s^3

A=6s^2

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

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

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

Now, solve for the length of the side of the cube to satisfy the problem condition, the rate of growth of the cube's volume is equal to $\frac{1}{16}$ times the rate of growth of its surface area:

$3s^2\frac{ds}{dt}=(\frac{1}{16})12s\frac{ds}{dt}$

$s=(\frac{1}{16})4=\frac{1}{4}$

The area of a face of the cube is given by:

a=s^2

$a=(\frac{1}{4})^2=\frac{1}{16}$

Report an Error

Example Question #483 : How To Find Rate Of Change

A cube is growing in size. What is the length of the diagonal of the cube at the time that the rate of growth of the cube's volume is equal to 9 times the rate of growth of its sides?

Possible Answers:

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

$3\sqrt{3}$

$9\sqrt{3}$

Correct answer:

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

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

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

Now, solve for the length of the side of the cube to satisfy the problem condition, the rate of growth of the cube's volume is equal to 9 times the rate of growth of its sides:

$3s^2\frac{ds}{dt}=(9)\frac{ds}{dt}$

$s^2=\frac{(9)}{3}$

$s=\sqrt{3}$

The diagonal is given by

$d=s\sqrt{3}$

d=3

Report an Error

Example Question #481 : Rate Of Change

A cube is growing in size. What is the surface area of the cube at the time that the rate of growth of the cube's volume is equal to 15 times the rate of growth of its sides?

Possible Answers:

$5\sqrt{6}$

$6\sqrt{5}$

Correct answer:

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

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

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

Now, solve for the length of the side of the cube to satisfy the problem condition, the rate of growth of the cube's volume is equal to 15 times the rate of growth of its sides:

$3s^2\frac{ds}{dt}=(15)\frac{ds}{dt}$

$s^2=\frac{(15)}{3}$

$s=\sqrt{5}$

The surface area is then:

A=6s^2

$A=6(\sqrt{5})^2=30$

Report an Error

Example Question #485 : How To Find Rate Of Change

A cube is growing in size. What is the length of the sides of the cube at the time that the rate of growth of the cube's volume is equal to 12 times the rate of growth of its sides?

Possible Answers:

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

$2\sqrt{3}$

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

Correct answer:

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

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

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

Now, solve for the length of the side of the cube to satisfy the problem condition, the rate of growth of the cube's volume is equal to 12 times the rate of growth of its sides:

$3s^2\frac{ds}{dt}=(12)\frac{ds}{dt}$

$s^2=\frac{(12)}{3}$

$s=\sqrt{4}=2$

Report an Error

Example Question #486 : How To Find Rate Of Change

A cube is growing in size. What is the area of a face of the cube at the time that the rate of growth of the cube's volume is equal to 3 times the rate of growth of its sides?

Possible Answers:

Correct answer:

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

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

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

Now, solve for the length of the side of the cube to satisfy the problem condition, the rate of growth of the cube's volume is equal to 3 times the rate of growth of its sides:

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

$s^2=\frac{(3)}{3}$

$s=\sqrt{1}=1$

The area of a face of the cube is then:

a=s^2

a=1

Report an Error

Example Question #487 : How To Find Rate Of Change

A cube is growing in size. What is the volume of the cube at the time that the rate of growth of the cube's volume is equal to 75 times the rate of growth of its sides?

Possible Answers:

125

$\frac{25}{9}$

$\frac{125}{27}$

Correct answer:

125

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

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

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

Now, solve for the length of the side of the cube to satisfy the problem condition, the rate of growth of the cube's volume is equal to 75 times the rate of growth of its sides:

$3s^2\frac{ds}{dt}=(75)\frac{ds}{dt}$

$s^2=\frac{(75)}{3}$

$s=\sqrt{25}=5$

Then to find the volume:

V=s^3

V=5^3=125

Report an Error

View Calculus Tutors

Chernyeh
Certified Tutor

Purdue University-Main Campus, Bachelor of Science, Business Administration and Management.

View Calculus Tutors

Shawn
Certified Tutor

Old Dominion University, Bachelor of Science, Mathematics. Old Dominion University, Master of Science, Computational Mathemat...

View Calculus Tutors

Elvis
Certified Tutor

Methodist University, Bachelor of Science, Mathematics and Computer Science.

All Calculus 1 Resources

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

Popular Subjects

Physics Tutors in Seattle, Calculus Tutors in Seattle, Computer Science Tutors in Houston, English Tutors in Seattle, Chemistry Tutors in Houston, French Tutors in Denver, GRE Tutors in Washington DC, MCAT Tutors in Seattle, LSAT Tutors in Boston, GMAT Tutors in Philadelphia

Popular Courses & Classes

GRE Courses & Classes in Phoenix, Spanish Courses & Classes in Los Angeles, SAT Courses & Classes in Phoenix, SSAT Courses & Classes in New York City, ACT Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Chicago, MCAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Atlanta, LSAT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Washington DC

Popular Test Prep

MCAT Test Prep in Houston, LSAT Test Prep in Los Angeles, ISEE Test Prep in Boston, MCAT Test Prep in Atlanta, SAT Test Prep in New York City, MCAT Test Prep in Washington DC, MCAT Test Prep in San Diego, SAT Test Prep in Miami, SSAT Test Prep in Atlanta, SSAT Test Prep in San Diego

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 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #562 : Rate

Example Question #563 : Rate

Example Question #564 : Rate

Example Question #481 : How To Find Rate Of Change

Example Question #482 : How To Find Rate Of Change

Example Question #483 : How To Find Rate Of Change

Example Question #481 : Rate Of Change

Example Question #485 : How To Find Rate Of Change

Example Question #486 : How To Find Rate Of Change

Example Question #487 : 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: