Call Now to Set Up Tutoring:

(888) 888-0446

ISEE Upper Level Quantitative : Spheres

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

Create An Account

All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Find The Radius Of A Sphere

The volume of a sphere is one cubic yard. Give its radius in inches.

Possible Answers:

$\frac{ 6\sqrt[3]{6 \pi^{2}} } {\pi} \textrm{ in}$

$\frac{ \sqrt[3]{6 \pi^{2}} } {2\pi}\textrm{ in}$

$\frac{ 18\sqrt{3 \pi } } {\pi} \textrm{ in}$

$\frac{ \sqrt{3 \pi } } {2 \pi} \textrm{ in}$

$\frac{ 18\sqrt[3]{6 \pi^{2}} } {\pi} \textrm{ in}$

Correct answer:

$\frac{ 18\sqrt[3]{6 \pi^{2}} } {\pi} \textrm{ in}$

Explanation:

The volume of a sphere with radius is

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

To find the radius in yards, we set V = 1 and solve for .

$1 = \frac{4}{3} \pi r^{3}$

$\frac{3}{4} \cdot 1 =\frac{3}{4} \cdot \frac{4}{3} \pi r^{3}$

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

$\frac{1}{\pi} \cdot \frac{3}{4} =\frac{1}{\pi} \cdot \pi r^{3}$

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

$r = \sqrt[3]{ \frac{3}{4\pi} } = \sqrt[3]{ \frac{3\cdot 2 \pi^{2}}{4\pi \cdot 2 \pi^{2}} }= \sqrt[3]{ \frac{6 \pi^{2}}{8 \pi^{3}} } = \frac{ \sqrt[3]{6 \pi^{2}} } {2\pi}$ yards.

Since the problem requests the radius in inches, multiply by 36:

$36 \times \frac{ \sqrt[3]{6 \pi^{2}} } {2\pi} = \frac{ 18\sqrt[3]{6 \pi^{2}} } {\pi}$

Report an Error

Example Question #1 : Spheres

In terms of $\pi$ , give the volume, in cubic feet, of a spherical tank with diameter 36 inches.

Possible Answers:

$\frac{9}{2} \pi \textrm{ ft}^{3}$

$9 \pi \textrm{ ft}^{3}$

$18 \pi \textrm{ ft}^{3}$

$3 \pi \textrm{ ft}^{3}$

$36 \pi \textrm{ ft}^{3}$

Correct answer:

$\frac{9}{2} \pi \textrm{ ft}^{3}$

Explanation:

36 inches = $36 \div 12 = 3$ feet, the diameter of the tank. Half of this, or $\frac{3}{2}$ feet, is the radius. Set $r = \frac{3}{2}$ , substitute in the volume formula, and solve for :

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

$V = \frac{4}{3} \pi\left \cdot \left( \frac{3}{2} \right ) ^{3}$

$V = \frac{4}{3}\cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \pi\left$

$V = \frac{1}{1}\cdot \frac{1}{1} \cdot \frac{3}{1} \cdot \frac{3}{2} \cdot \pi\left$

$V = \frac{9}{2} \pi$

Report an Error

Example Question #2 : Spheres

Which is the greater quantity?

(a) The volume of a sphere with radius

(b) The volume of a cube with sidelength

Possible Answers:

It is impossible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

A sphere with radius has diameter and can be inscribed inside a cube of sidelength . Therefore, the cube in (b) has the greater volume.

Report an Error

Example Question #3 : Spheres

Which is the greater quantity?

(a) The volume of a cube with sidelength inches.

(b) The volume of a sphere with radius inches.

Possible Answers:

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

You do not need to calculate the volumes of the figures. All you need to do is observe that a sphere with radius inches has diameter inches, and can therefore be inscribed inside the cube with sidelength inches. This give the cube larger volume, making (a) the greater quantity.

Report an Error

Example Question #3 : Spheres

Which is the greater quantity?

(a) The volume of a sphere with diameter one foot

(b) $864 \textrm{ in}^{3}$

Possible Answers:

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

Correct answer:

(a) is greater.

Explanation:

The radius of the sphere is one half of its diameter of one foot, which is six inches, so substitute r = 6 :

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

$V = \frac{4}{3} \pi \cdot 6 ^{3}$

$V = \frac{4}{3} \pi \cdot 216$

$V = 288 \pi \approx 288 \cdot 3.14 = 904.32$ cubic inches,

which is greater than $864 \textrm{ in}^{3}$ .

Report an Error

Example Question #5 : Spheres

is a positive number. Which is the greater quantity?

(A) The volume of a cube with edges of length

(B) The volume of a sphere with radius

Possible Answers:

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(A) is greater

(B) is greater

Correct answer:

(A) is greater

Explanation:

No calculation is really needed here, as a sphere with radius - and, subsequently, diameter - can be inscribed inside a cube of sidelength . This makes (A), the volume of the cube, the greater.

Report an Error

Example Question #6 : Spheres

Which is the greater quantity?

(a) The radius of a sphere with surface area $36 \pi$

(b) The radius of a sphere with volume $36 \pi$

Possible Answers:

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

Correct answer:

(a) and (b) are equal

Explanation:

The formula for the surface area of a sphere, given its radius , is

$A = 4 \pi r^{2}$

The sphere in (a) has surface area $36 \pi$ , so

$36 \pi = 4 \pi r^{2}$

$36 \pi\div 4 \pi = 4 \pi r^{2} \div 4 \pi$

$9 = r^{2}$

$r = \sqrt{9} = 3$

The formula for the volume of a sphere, given its radius , is

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

The sphere in (b) has volume $36 \pi$ , so

$36 \pi = \frac{4}{3} \pi r^{3}$

$\frac{3} {4} \cdot 36 \pi = \frac{3} {4} \cdot \frac{4}{3} \pi r^{3}$

$27 \pi = \pi r^{3}$

$27 \pi\div \pi = \pi r^{3} \div \pi$

$27 = r^{3}$

$r = \sqrt[3]{27} = 3$

The radius of both spheres is 3.

Report an Error

Example Question #6 : Spheres

In terms of $\pi$ , give the surface area, in square inches, of a spherical water tank with a diameter of 20 feet.

Possible Answers:

$2,304,000 \pi \textrm{ in}^{2}$

$57,600 \pi\textrm{ in}^{2}$

$307,200\pi\textrm{ in}^{2}$

$18,432,000\pi\textrm{ in}^{2}$

$230,400 \pi\textrm{ in}^{2}$

Correct answer:

$57,600 \pi\textrm{ in}^{2}$

Explanation:

feet = $20 \times 12 = 240$ inches, the diameter of the tank; half of this, or 120 inches, is the radius. Set r = 120 , substitute in the surface area formula, and solve for :

$A =4\pi r^{2}$

$A =4\pi \cdot 120^{2}$

$A =4\pi \cdot 14,400$

$A = 57,600 \pi$

Report an Error

Example Question #4 : Spheres

Which is the greater quantity?

(a) The surface area of a sphere with radius 1

(b) 12

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(a) is greater.

Explanation:

The surface area of a sphere can be found using the formula

$A = 4\pi r^{2}$ .

The surface area of the given sphere can be found by substituting r = 1 :

$A = 4\pi \cdot 1^{2} = 4\pi$

$\pi > 3$ so $4 \times \pi >4 \times 3$ , or $4\pi > 12$

This makes (a) greater.

Report an Error

Example Question #5 : Spheres

Sphere A has volume $972 \pi \textrm{ in}^{3}$ . Sphere B has surface area $400 \pi \textrm{ in}^{2}$ . Which is the greater quantity?

(a) The radius of Sphere A

(b) The radius of Sphere B

Possible Answers:

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Correct answer:

(b) is greater

Explanation:

(a) Substitute $V = 972 \pi$ in the formula for the volume of a sphere:

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

$\frac{4}{3}\pi r^{3}= 972 \pi$

$\frac{4}{3}\pi r^{3} \div \frac{4}{3}\pi = 972 \pi \div \frac{4}{3}\pi$

$\frac{4}{3}\pi r^{3} \cdot \frac{3}{4\pi} = 972 \pi\cdot \frac{3}{4\pi}$

$r^{3} = 729$

$r =\sqrt[3]{ 729} = 9$ inches

(b) Substitute $A = 400 \pi$ in the formula for the surface area of a sphere:

$4\pi r^{2}= A$

$4\pi r^{2}= 400 \pi$

$4\pi r^{2} \div 4\pi = 400 \pi \div 4\pi$

$r^{2} = 100$

$r = \sqrt{100} = 10$ inches

(b) is greater.

Report an Error

View Tutors

Gabby
Certified Tutor

University of Oregon, Bachelor in Arts, Journalism. University of Oregon, Current Grad Student, Communication, General.

View Tutors

Eva
Certified Tutor

Technicka Univerzita Liberec, Bachelor of Education, Education.

View Tutors

Gregory
Certified Tutor

Pennsylvania State University-Main Campus, Bachelor of Science, Chemical Engineering. Carnegie Mellon University, Doctor of P...

All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

ISEE Tutors in Washington DC, Computer Science Tutors in Phoenix, Chemistry Tutors in New York City, Chemistry Tutors in Miami, LSAT Tutors in Chicago, SAT Tutors in Philadelphia, ACT Tutors in San Diego, LSAT Tutors in Los Angeles, GMAT Tutors in Miami, ISEE Tutors in Dallas Fort Worth

Popular Courses & Classes

MCAT Courses & Classes in Chicago, SSAT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Boston, ISEE Courses & Classes in Philadelphia, LSAT Courses & Classes in Washington DC, ISEE Courses & Classes in Chicago, MCAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Los Angeles, GRE Courses & Classes in Miami, SAT Courses & Classes in Miami

Popular Test Prep

GMAT Test Prep in Philadelphia, MCAT Test Prep in Denver, GMAT Test Prep in Seattle, GRE Test Prep in Houston, SAT Test Prep in New York City, GRE Test Prep in Chicago, SAT Test Prep in Los Angeles, SSAT Test Prep in Washington DC, ISEE Test Prep in Boston, GRE Test Prep in Miami

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

ISEE Upper Level Quantitative : Spheres

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : How To Find The Radius Of A Sphere

Example Question #1 : Spheres

Example Question #2 : Spheres

Example Question #3 : Spheres

Example Question #3 : Spheres

Example Question #5 : Spheres

Example Question #6 : Spheres

Example Question #6 : Spheres

Example Question #4 : Spheres

Example Question #5 : Spheres

All ISEE Upper Level Quantitative Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: