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ISEE Upper Level Quantitative : How to find the length of an edge

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

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Example Questions

Example Question #1 : Prisms

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:

$27 \textrm{ in}$ $\displaystyle 27 \textrm{ in}$

$18 \textrm{ in}$ $\displaystyle 18 \textrm{ in}$

$30\textrm{ in}$ $\displaystyle 30\textrm{ in}$

$36\textrm{ in}$ $\displaystyle 36\textrm{ in}$

$24 \textrm{ in}$ $\displaystyle 24 \textrm{ in}$

Correct answer:

$24 \textrm{ in}$ $\displaystyle 24 \textrm{ in}$

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

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

The volume of a rectangular prism is

V = LHW $\displaystyle V = LHW$

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

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

$5,832= 243 \cdot W$ $\displaystyle 5,832= 243 \cdot W$

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

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:

$24\textrm{ in}$ $\displaystyle 24\textrm{ in}$

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

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

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

$18\textrm{ in}$ $\displaystyle 18\textrm{ in}$

Correct answer:

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

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

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

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

$\displaystyle A = 2LW + 2LH + 2 WH$.

So, with substitutiton, we can find the width:

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

$\displaystyle 1,944= 54W +486+ 18 W$

$\displaystyle 72W +486 = 1,944$

$\displaystyle 72W +486 - 486 = 1,944 - 486$

$\displaystyle 72W = 1,458$

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

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

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Example Question #2 : Prisms

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

Possible Answers:

$\frac{192}{t}$ $\displaystyle \frac{192}{t}$

16t $\displaystyle 16t$

$\frac{96}{t}$ $\displaystyle \frac{96}{t}$

96t $\displaystyle 96t$

$\frac{16}{t}$ $\displaystyle \frac{16}{t}$

Correct answer:

$\frac{192}{t}$ $\displaystyle \frac{192}{t}$

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 $12^{3} = 1,728$ $\displaystyle 12^{3} = 1,728$ cubic inches.

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

We can solve for $\displaystyle H$ as follows:

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

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

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

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ISEE Upper Level Quantitative : How to find the length of an edge

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : Prisms

Example Question #2 : Solid Geometry

Example Question #2 : Prisms

All ISEE Upper Level Quantitative Resources

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