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ISEE Upper Level Quantitative : Prisms

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Example Questions

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:

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

(A) is greater

(B) 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.

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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:

$27 \textrm{ in}$

$24 \textrm{ in}$

$36\textrm{ in}$

$30\textrm{ in}$

$18 \textrm{ in}$

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.

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Example Question #2 : Solid Geometry

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:

$24\textrm{ in}$

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

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

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

$18\textrm{ in}$

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

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Example Question #2 : Solid Geometry

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:

$\frac{16}{t}$

$\frac{192}{t}$

$\frac{96}{t}$

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:

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

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

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

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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:

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

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

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

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

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

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

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

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

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

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Example Question #2 : 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:

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

(B) is greater

(A) and (B) are equal

(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

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Prisms

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 #2 : Solid Geometry

Example Question #2 : Solid Geometry

Example Question #1 : Solid Geometry

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

All ISEE Upper Level Quantitative Resources

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