Call Now to Set Up Tutoring:

(888) 888-0446

ISEE Upper Level Quantitative : Solid Geometry

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

← Previous 1 2 3 4 5 6 Next →

Example Question #1 : How To Find The Surface Area Of A Prism

is a positive number. Which is the greater quantity?

(A) The surface area of a rectangular prism with length 16t , width , and height

(B) The surface area of a rectangular prism with length , width , and height .

Possible Answers:

(B) is greater

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

(A) is greater

(A) and (B) are equal

Correct answer:

(A) is greater

Explanation:

The surface area of a rectangular prism can be determined using the formula:

S = 2(LW+WH +LH)

Using substitutions, the surface areas of the prisms can be found as follows:

The prism in (A):

$S = 2( 16t \cdot 9t +9t \cdot t +16t \cdot t)$

$S = 2( 144t ^{2}+9t^{2} +16t^{2})$

$S = 2( 169t ^{2})$

$S = 338t ^{2}$

$S = 2( 8t \cdot 6t +6t \cdot 3t +8t \cdot 3 t)$

$S = 2( 48t^{2} +18t ^{2}+24 t^{2})$

$S = 2( 90 t^{2})$

$S = 180 t^{2}$

Regardless of the value of , $338t ^{2} > 180t ^{2}$ - that is, the first prism has the greater surface area. (A) is greater.

Report an Error

Example Question #1 : How To Find The Length Of An Edge

A cube has sidelength one and one-half feet; a rectangular prism of equal volume has length 27 inches and height 9 inches. Give the width of the prism in inches.

Possible Answers:

$36\textrm{ in}$

$30\textrm{ in}$

$24 \textrm{ in}$

$18 \textrm{ in}$

$27 \textrm{ in}$

Correct answer:

$24 \textrm{ in}$

Explanation:

One and one half feet is equal to eighteen inches, so the volume of the cube, in cubic inches, is the cube of this, or

$V = 18^{3} = 5,832$ cubic inches.

The volume of a rectangular prism is

V = LHW

Since its volume is the same as that of the cube, and its length and height are 27 and 9 inches, respectively, we can rewrite this as

$5,832= 27 \cdot 9 \cdot W$

$5,832= 243 \cdot W$

$W = 5,832 \div 243 = 24$

The width is 24 inches.

Report an Error

Example Question #1 : How To Find The Length Of An Edge

A cube has sidelength one and one-half feet; a rectangular prism of equal surface area has length 27 inches and height 9 inches. Give the width of the prism in inches.

Possible Answers:

$22 \frac{1}{2} \textrm{ in}$

$20 \frac{1}{4} \textrm{ in}$

$24\textrm{ in}$

$27\frac{1}{2} \textrm{ in}$

$18\textrm{ in}$

Correct answer:

$20 \frac{1}{4} \textrm{ in}$

Explanation:

One and one half feet is equal to eighteen inches, so the surface area of the cube, in square inches, is six times the square of this, or

$6 \cdot 18^{2} = 6 \cdot 324 = 1,944$ square inches.

The surface area of a rectangular prism is determined by the formula

A = 2LW + 2LH + 2 WH .

So, with substitutiton, we can find the width:

$1,944= 2 \cdot 27 \cdot W + 2\cdot 27 \cdot 9 + 2 \cdot W \cdot 9$

1,944= 54W +486+ 18 W

72W +486 = 1,944

72W +486 - 486 = 1,944 - 486

72W = 1,458

$72W\div 72 = 1,458 \div 72$

$W = 20 \frac{1}{4}$ inches

Report an Error

Example Question #3 : How To Find The Length Of An Edge

A rectangular prism has volume one cubic foot; its length and width are, respectively, 9 inches and inches. Which of the following represents the height of the prism in inches?

Possible Answers:

$\frac{192}{t}$

$\frac{96}{t}$

$\frac{16}{t}$

16t

96t

Correct answer:

$\frac{192}{t}$

Explanation:

The volume of a rectangular prism is the product of its length, its width, and its height. The prism's volume of one cubic foot is equal to $12^{3} = 1,728$ cubic inches.

Therefore, LWH = V can be rewritten as $9 \cdot t \cdot H = 1,728$ .

We can solve for as follows:

$\frac{9 t \cdot H }{9t}= \frac{1,728 }{9t}$

$H = \frac{1,728 \div 9 }{9t \div 9 }$

$H = \frac{192}{t}$

Report an Error

Example Question #1 : Solid Geometry

A large crate in the shape of a rectangular prism has dimensions 5 feet by 4 feet by 12 feet. Give its volume in cubic yards.

Possible Answers:

$90\textrm{ yd}^{3}$

$10 \textrm{ yd}^{3}$

$80 \textrm{ yd}^{3}$

$7 \frac{2}{9 } \textrm{ yd}^{3}$

$8 \frac{8}{9 } \textrm{ yd}^{3}$

Correct answer:

$8 \frac{8}{9 } \textrm{ yd}^{3}$

Explanation:

Divide each dimension by 3 to convert feet to yards, then multiply the three dimensions together:

V = LWH

$V = \frac{5}{3} \cdot \frac{4}{3} \cdot \frac{12}{3}$

$V = \frac{5}{3} \cdot \frac{4}{3} \cdot \frac{4}{1}$

$V = \frac{80}{9 } = 8 \frac{8}{9 } \textrm{ yd}^{3}$

Report an Error

Example Question #1 : Find The Volume Of A Right Rectangular Prism With Fractional Edge Lengths: Ccss.Math.Content.6.G.A.2

Which is the greater quantity?

(A) The volume of a rectangular solid ten inches by twenty inches by fifteen inches

(B) The volume of a cube with sidelength sixteen inches

Possible Answers:

(A) and (B) are equal

(B) is greater

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

(A) is greater

Correct answer:

(B) is greater

Explanation:

The volume of a rectangular solid ten inches by twenty inches by fifteen inches is

$V = 10 \cdot 20 \cdot15 = 3,000$ cubic inches.

The volume of a cube with sidelength 13 inches is

$V = 16^{3} = 16 \cdot 16 \cdot 16 = 4,096$ cubic inches.

This makes (B) greater

Report an Error

Example Question #1 : How To Find The Volume Of A Pyramid

Pyramid 1 has a square base with sidelength ; its height is .

Pyramid 2 has a square base with sidelength ; its height is .

Which is the greater quantity?

(a) The volume of Pyramid 1

(b) The volume of Pyramid 2

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:

Use the formula $V = \frac {1}{3} Bh$ on each pyramid.

(a) $B = x^{2}; h = 2x$

$V = \frac {1}{3} Bh = \frac {1}{3} \cdot x^{2} \cdot 2x = \frac {2}{3} x^{3}$

(b) $B = (2 x)^{2} = 4x^{2}; h = x$

$V = \frac {1}{3} Bh = \frac {1}{3} \cdot 4x^{2} \cdot x = \frac {4}{3} x^{3}$

Regardless of , (b) is the greater quantity.

Report an Error

Example Question #2 : How To Find The Volume Of A Pyramid

Which is the greater quantity?

(a) The volume of a pyramid with height 4, the base of which has sidelength 1

(b) The volume of a pyramid with height 1, the base of which has sidelength 2

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

The volume of a pyramid with height and a square base with sidelength is

$V = \frac{1}{3} Bh = \frac{1}{3} s^{2}h$ .

(a) Substitute s=1,h=4 : $V = \frac{1}{3} s^{2}h = \frac{1}{3} \cdot 1^{2} \cdot 4 = \frac{4}{3}$

(b) Substitute s=2,h=1 : $V = \frac{1}{3} s^{2}h = \frac{1}{3} \cdot 2^{2} \cdot 1 = \frac{4}{3}$

The two pyramids have equal volume.

Report an Error

Example Question #1 : How To Find The Volume Of A Pyramid

Which is the greater quantity?

(a) The volume of a pyramid whose base is a square with sidelength 8 inches

(b) The volume of a pyramid whose base is an equilateral triangle with sidelength one foot

Possible Answers:

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(a) is greater.

Correct answer:

It is impossible to tell from the information given.

Explanation:

The volume of a pyramid is one-third of the product of the height and the area of the base. The areas of the bases can be calculated, but no information is given about the heights of the pyramids. There is not enough information to determine which one has the greater volume.

Report an Error

Example Question #4 : How To Find The Volume Of A Pyramid

A pyramid with a square base has height equal to the perimeter of its base. Its volume is . In terms of , what is the length of each side of its base?

Possible Answers:

$\sqrt[3]{\frac{3V}{4}}$

$\frac{\sqrt[3]{3V }}{8}$

$\frac{\sqrt[3]{3V }}{2}$

$\sqrt[3]{\frac{3V}{2}}$

$\sqrt[3]{3V }$

Correct answer:

$\sqrt[3]{\frac{3V}{4}}$

Explanation:

The volume of a pyramid is given by the formula

$V = \frac{1}{3} A h$

where is the area of its base and is its height.

Let be the length of one side of the square base. Then the height is equal to the perimeter of that square, so

h = 4s

and the area of the base is

$A = s^{2}$

So the volume formula becomes

$\frac{1}{3} \cdot s^{2} \cdot 4s = V$

Solve for :

$\frac{4}{3} \cdot s^{3} = V$

$\frac{3}{4} \cdot \frac{4}{3} \cdot s^{3} =\frac{3}{4} \cdot V$

$s^{3} =\frac{3V}{4}$

$s =\sqrt[3]{\frac{3V}{4}}$

Report an Error

← Previous 1 2 3 4 5 6 Next →

View Tutors

Jeromy
Certified Tutor

Washington State University, Bachelor of Science, Psychology. University of Washington-Seattle Campus, Master of Arts Teachin...

View Tutors

Robert
Certified Tutor

University of North Carolina at Chapel Hill, Bachelor in Arts, English. North Carolina Central University, Master of Arts, En...

View Tutors

Caroline
Certified Tutor

University of Florida, Bachelor of Science, Psychology.

All ISEE Upper Level Quantitative Resources

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

Popular Subjects

SSAT Tutors in Dallas Fort Worth, Math Tutors in Houston, SSAT Tutors in San Diego, Statistics Tutors in Washington DC, Biology Tutors in Atlanta, Computer Science Tutors in Dallas Fort Worth, GMAT Tutors in Houston, Biology Tutors in Miami, English Tutors in Atlanta, Spanish Tutors in San Diego

Popular Courses & Classes

Spanish Courses & Classes in New York City, ACT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Washington DC, ISEE Courses & Classes in Atlanta, GMAT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Washington DC, LSAT Courses & Classes in Houston, ISEE Courses & Classes in Philadelphia, SSAT Courses & Classes in San Diego

Popular Test Prep

LSAT Test Prep in Denver, SSAT Test Prep in San Francisco-Bay Area, ACT Test Prep in Dallas Fort Worth, ACT Test Prep in Miami, ACT Test Prep in Phoenix, LSAT Test Prep in Phoenix, GRE Test Prep in Seattle, MCAT Test Prep in Phoenix, ISEE Test Prep in Los Angeles, SAT Test Prep in Dallas Fort Worth

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 : Solid Geometry

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 Surface Area Of A Prism

Example Question #1 : How To Find The Length Of An Edge

Example Question #1 : How To Find The Length Of An Edge

Example Question #3 : How To Find The Length Of An Edge

Example Question #1 : Solid Geometry

Example Question #1 : Find The Volume Of A Right Rectangular Prism With Fractional Edge Lengths: Ccss.Math.Content.6.G.A.2

Example Question #1 : How To Find The Volume Of A Pyramid

Example Question #2 : How To Find The Volume Of A Pyramid

Example Question #1 : How To Find The Volume Of A Pyramid

Example Question #4 : How To Find The Volume Of A Pyramid

All ISEE Upper Level Quantitative Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: