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ISEE Upper Level Math : Geometry

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6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Volume Of A Rectangular Solid

If a cube is inches tall, what is its volume?

Possible Answers:

512

384

500

Not enough information provided.

Correct answer:

512

Explanation:

To find the volume of a cube, we multiply length by width by height, which can be represented with the forumla $v=l\cdot w\cdot h$ . Since a cube has equal sides, we can use for all three values.

$8\cdot 8\cdot 8=512\: in ^{3}$

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Example Question #101 : Geometry

What is the volume of a cube with a side length equal to inches?

Possible Answers:

$53\: in^{3}$

$49\: in^{3}$

$343\: in^{3}$

$21\: in^{3}$

Correct answer:

$343\: in^{3}$

Explanation:

The volume of a a cube (or rectangular prism) can be solved using the following equation:

$V=l\times\ w\times\ h$

$V=7in\times 7in\times 7in=343in^3$

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

Give the volume of a cube with surface area 150 square inches.

Possible Answers:

$250\textup{ in}^{3}$

$250\sqrt{3}\textup{ in}^{3}$

$375\textup{ in}^{3}$

$125\textup{ in}^{3}$

$125\sqrt{3}\textup{ in}^{3}$

Correct answer:

$125\textup{ in}^{3}$

Explanation:

Let be the length of one edge of the cube. Since its surface area is 150 square inches, one face has one-sixth of this area, or $\frac{150}{6} = 25$ square inches. Therefore, $s^{2} = 25$ , and s = 5 .

The volume is the cube of this, or $V= 5^{3}=125$ cubic inches.

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Example Question #12 : Solid Geometry

What is the volume of a cube in which the edge is equal to $\sqrt[3]{x}$ , and the value of is:

$x=5(4+3^{2})-3$

Possible Answers:

Correct answer:

Explanation:

First, the value of x must be solved for:

$x=5(4+3^{2})-3$

x=5(4+9)-3

x=5(13)-3

x=62

Given the edge of the cube is $\sqrt[3]{x}$ , plugging in the value of x results in $\sqrt[3]{62}$ . Thus, the area would be equal to this value cubed, which would result in 62.

Thus, 62 is the correct answer.

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Example Question #13 : Solid Geometry

The length of a diagonal of one face of a cube is . Give the volume of the cube.

Possible Answers:

$\frac{N^{3} \sqrt{3} }{3}$

$\frac{N^{3} \sqrt{2} }{4}$

$3N^{3}$

$\frac{N^{3} \sqrt{2} }{2}$

$\frac{N^{3} \sqrt{3} }{9}$

Correct answer:

$\frac{N^{3} \sqrt{2} }{4}$

Explanation:

Since a diagonal of a square face of the cube is, each side of each square has length $\frac{N}{\sqrt{2}}=\frac{N\sqrt{2}}{2}$ .

Cube this to get the volume of the cube:

$\left (\frac{N\sqrt{2}}{2} \right )^{3}$

$= \frac{N^{3}\left ( \sqrt{2} \right )^{3} }{2^{3} }$

$= \frac{N^{3} \cdot 2 \sqrt{2} }{8}$

$= \frac{N^{3} \sqrt{2} }{4}$

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

The distance from one vertex of a perfectly cubic aquarium to its opposite vertex is 1.5 meters. Give the volume of the aquarium in liters.

1 cubic meter = 1,000 liters.

Possible Answers:

The correct answer is not given among the other responses.

$843\frac{3}{4} \textup{ L}$

$750\textup{ L}$

$375 \textup{ L}$

$421\frac{7}{8} \textup{ L}$

Correct answer:

The correct answer is not given among the other responses.

Explanation:

Let be the length of one edge of the cube. By the three-dimensional extension of the Pythagorean Theorem,

$s^{2}+s^{2}+s^{2}=1.5^{2}$

$3s^{2}=2.25$

$s^{2} = 0.75 = \frac{3}{4}$

$s = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{\sqrt{4}}} = \frac{\sqrt{3}}{2}$ meters.

Cube this sidelength to get the volume:

$s ^{3 } =\left ( \frac{\sqrt{3}}{2} \right )^{3 }= \frac{\left (\sqrt{3}\right )^{3 }}{2^{3 } } = \frac{ 3\sqrt{3} }{8 }$ cubic meters.

To convert this to liters, multiply by 1,000:

$\frac{ 3\sqrt{3} }{8 } \cdot 1,000 = \frac{ 3\sqrt{3} }{8 } \cdot \frac{1,000}{1} = \frac{ 3\sqrt{3} }{1 } \cdot \frac{125}{1}= 375\sqrt{3}$ liters.

This is not among the given responses.

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Example Question #1 : Finding Volume Of A Cube

Give the volume of a cube with surface area 240 square inches.

Possible Answers:

$120 \sqrt{10} \textup{ in}^{3}$

$120 \textup{ in}^{3}$

$80 \sqrt{10} \textup{ in}^{3}$

$80 \textup{ in}^{3}$

$40 \sqrt{10} \textup{ in}^{3}$

Correct answer:

$80 \sqrt{10} \textup{ in}^{3}$

Explanation:

Let be the length of one edge of the cube. Since its surface area is 240 square inches, one face has one-sixth of this area, or $\frac{240}{6} = 40$ square inches. Therefore, $s^{2} = 40$ , and $s = \sqrt{40} = 2\sqrt{10}$ .

The volume is the cube of this, or $V= \left ( 2\sqrt{10} \right )^{3}= 2^{3} \cdot \left ( \sqrt{10} \right )^{3}= 8 \cdot 10 \sqrt{10}= 80 \sqrt{10}$ cubic inches.

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Example Question #15 : Volume Of A Rectangular Solid

An aquarium is shaped like a perfect cube; the perimeter of each glass face is 8.4 meters. If it is filled to the recommended $90\%$ capacity, then, to the nearest hundred cubic liters, how much water will it contain?

Possible Answers:

$9\textup,300\textup{ L}$

$8\textup,300\textup{ L}$

$4\textup,000\textup{ L}$

Insufficient information is given to answer the question.

$4\textup,400\textup{ L}$

Note: $1\textup{ cubic meter}= 1\textup{,}000\textup{ cubic liters}$

Correct answer:

$8\textup,300\textup{ L}$

Explanation:

A perfect cube has square faces; if a face has perimeter 8.4 meters, then each side of each face measures one fourth of this, or 2.1 meters. The volume of the tank is the cube of this, or

$2.1^{3} = 9.261$ cubic meters.

Its capacity in liters is $9.261 \times 1,000 = 9,261$ liters.

$90\%$ of this is

$9,261 \times 0.9 = 8,335$ liters.

This rounds to $8\textup,300$ liters, the correct response.

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Example Question #331 : Geometry

Your friend gives you a puzzle cube for your birthday. If the length of one edge is 5cm, what is the volume of the cube?

Possible Answers:

125cm^3

25cm^3

15cm^3

625cm^3

Correct answer:

125cm^3

Explanation:

Your friend gives you a puzzle cube for your birthday. If the length of one edge is 5cm, what is the volume of the cube?

To find the volume of a cube, use the following formula:

$V_{cube}=s^3$

Where s is the side length.

Plug in what we know to get our answer:

V=(5cm)^3=125cm^3

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Example Question #4 : Cubes

A cube has a side length of 9 mm , what is the volume of the cube?

Possible Answers:

486mm^3

81mm^3

72.9mm^3

729mm^3

Correct answer:

729mm^3

Explanation:

A cube has a side length of 9 mm , what is the volume of the cube?

To find the volume of a cube, use the following formula:

$V_{cube}=s^3$

Plug in our known side length and solve

$V_{cube}=(9mm)^3=729mm^3$

Making our answer:

729mm^3

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Geometry

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : Volume Of A Rectangular Solid

Example Question #101 : Geometry

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

Example Question #12 : Solid Geometry

Example Question #13 : Solid Geometry

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

Example Question #1 : Finding Volume Of A Cube

Example Question #15 : Volume Of A Rectangular Solid

Example Question #331 : Geometry

Example Question #4 : Cubes

All ISEE Upper Level Math Resources

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