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 #3534 : Calculus

Find the slope at x=7 given the following function:

$f(x)= e^{ln(x)}$

Possible Answers:

Answer not listed

Correct answer:

Explanation:

In order to find the slope of a certain point given a function, you must first find the derivative.

The function simplifies to: $f(x)= e^{ln(x)} =x$

In this case the derivative is: f'(x)= 1

Then plug x=7 into the derivative function: f'(7)= 1

Therefore, the slope is:

Report an Error

Example Question #3535 : Calculus

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 112 times the rate of growth of its surface area?

Possible Answers:

448

112

224

Correct answer:

448

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 112 times the rate of growth of its surface area:

$3s^2\frac{ds}{dt}=(112)12s\frac{ds}{dt}$

s=(112)4

s=448

Report an Error

Example Question #3536 : Calculus

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 127 times the rate of growth of its surface area?

Possible Answers:

254

$508\sqrt{3}$

$127\sqrt{3}$

$254\sqrt{3}$

508

Correct answer:

$508\sqrt{3}$

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 127 times the rate of growth of its surface area:

$3s^2\frac{ds}{dt}=(127)12s\frac{ds}{dt}$

s=(127)4=508

The diagonal is then:

$d=s\sqrt{3}$

$d=508\sqrt{3}$

Report an Error

Example Question #3537 : Calculus

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 95 times the rate of growth of its surface area?

Possible Answers:

1732800

1299600

288800

866400

144400

Correct answer:

866400

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 95 times the rate of growth of its surface area:

$3s^2\frac{ds}{dt}=(95)12s\frac{ds}{dt}$

s=(95)4=380

The surface area is then:

A=6s^2

A=6(380)^2=866400

Report an Error

Example Question #3538 : Calculus

A cube is diminishing in size. What is the length of the sides of the cube at the time that the rate of shrinkage of the cube's volume is equal to $\frac{9}{10}$ times the rate of shrinkage of its surface area?

Possible Answers:

$\frac{9}{5}$

$\frac{3}{5}$

$\frac{36}{5}$

$\frac{18}{5}$

$\frac{72}{5}$

Correct answer:

$\frac{18}{5}$

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 shrinkage of the cube's volume is equal to $\frac{9}{10}$ times the rate of shrinkage of its surface area:

$3s^2\frac{ds}{dt}=(\frac{9}{10})12s\frac{ds}{dt}$

$s=(\frac{9}{10})4=\frac{18}{5}$

Report an Error

Example Question #2505 : Functions

A cube is diminishing in size. What is volume of the cube at the time that the rate of shrinkage of the cube's volume is equal to $\frac{14}{15}$ times the rate of shrinkage of its surface area?

Possible Answers:

$\frac{97321}{625}$

$\frac{3136}{225}$

$\frac{175616}{3375}$

$\frac{108}{25}$

$\frac{9834496}{50625}$

Correct answer:

$\frac{175616}{3375}$

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 shrinkage of the cube's volume is equal to $\frac{14}{15}$ times the rate of shrinkage of its surface area

$3s^2\frac{ds}{dt}=(\frac{14}{15})12s\frac{ds}{dt}$

$s=(\frac{14}{15})4=\frac{56}{15}$

The volume is then:

V=s^3

$V=(\frac{56}{15})^3=\frac{175616}{3375}$

Report an Error

Example Question #2511 : Functions

A cube is diminishing in size. What is the surface area of the cube at the time that the rate of shrinkage of the cube's volume is equal to $\frac{5}{6}$ times the rate of shrinkage of its surface area?

Possible Answers:

$\frac{400}{3}$

$\frac{400}{9}$

$\frac{100}{3}$

$\frac{200}{9}$

$\frac{200}{3}$

Correct answer:

$\frac{200}{3}$

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 shrinkage of the cube's volume is equal to $\frac{5}{6}$ times the rate of shrinkage of its surface area

$3s^2\frac{ds}{dt}=(\frac{5}{6})12s\frac{ds}{dt}$

$s=(\frac{5}{6})4=\frac{10}{3}$

The surface area at this time is then:

A=6s^2

$A=6(\frac{10}{3})^2=\frac{200}{3}$

Report an Error

Example Question #2512 : Functions

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 surface area when its sides have length 1164?

Possible Answers:

291

485

388

194

Correct answer:

291

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

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

$\phi=\frac{s}{4}$

$\phi =\frac{1164}{4}=291$

Report an Error

Example Question #2513 : Functions

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 surface area when its sides have length 620?

Possible Answers:

310

186

155

930

Correct answer:

155

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

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

$\phi=\frac{s}{4}$

$\phi =\frac{620}{4}=155$

Report an Error

Example Question #2514 : Functions

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 surface area when its sides have length 1656?

Possible Answers:

6624

207

552

414

Correct answer:

414

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

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

$\phi=\frac{s}{4}$

$\phi =\frac{1656}{4}=414$

Report an Error

View Calculus Tutors

Usman
Certified Tutor

New York University, Bachelor in Arts, Political Science and Government.

View Calculus Tutors

Jeremy
Certified Tutor

University of Illinois at Chicago, Bachelor of Science, Physics.

View Calculus Tutors

Bobby
Certified Tutor

University of Houston - Downtown, Bachelor of Science, Applied Mathematics.

All Calculus 1 Resources

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

Popular Subjects

GMAT Tutors in San Diego, Computer Science Tutors in Phoenix, Physics Tutors in San Francisco-Bay Area, Chemistry Tutors in New York City, Biology Tutors in Denver, SAT Tutors in Denver, Statistics Tutors in San Diego, SSAT Tutors in New York City, ACT Tutors in Seattle, Calculus Tutors in Miami

Popular Courses & Classes

ACT Courses & Classes in Denver, SAT Courses & Classes in Houston, GRE Courses & Classes in Atlanta, SSAT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Miami, ACT Courses & Classes in New York City, ACT Courses & Classes in Miami, ACT Courses & Classes in Washington DC, SSAT Courses & Classes in Boston, GMAT Courses & Classes in New York City

Popular Test Prep

GMAT Test Prep in New York City, GMAT Test Prep in Philadelphia, ACT Test Prep in Washington DC, MCAT Test Prep in Chicago, GRE Test Prep in Atlanta, ACT Test Prep in Phoenix, LSAT Test Prep in Washington DC, GRE Test Prep in Miami, ACT Test Prep in San Diego, SSAT Test Prep in San Francisco-Bay Area

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 #3534 : Calculus

Example Question #3535 : Calculus

Example Question #3536 : Calculus

Example Question #3537 : Calculus

Example Question #3538 : Calculus

Example Question #2505 : Functions

Example Question #2511 : Functions

Example Question #2512 : Functions

Example Question #2513 : Functions

Example Question #2514 : Functions

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: