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SAT Math : Solid Geometry

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Example Questions

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

A cube has 2 faces painted red and the remaining faces painted green. The total area of the green faces is 36 square inches. What is the volume of the cube in cubic inches?

Possible Answers:

Correct answer:

Explanation:

Cubes have 6 faces. If 2 are red, then 4 must be green. We are told that the total area of the green faces is 36 square inches, so we divide the total area of the green faces by the number of green faces (which is 4) to get the area of each green face: 36/4 = 9 square inches. Since each of the 6 faces of a cube have the same size, we know that each edge of the cube is √9 = 3 inches. Therefore the volume of the cube is 3 in x 3 in x 3 in = 27 cubic inches.

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

If a waterproof box is 50cm in length, 20cm in depth, and 30cm in height, how much water will overflow if 35 liters of water are poured into the box?

Possible Answers:

15 liters

1 liters

30 lites

5 liters

No water will flow out of the box

Correct answer:

5 liters

Explanation:

The volume of the box is 50 * 20 * 30 cm = 30,000 cm³.

1cm³ = 1mL, 30,000 cm³ = 30,000mL = 30 L.

Because the volume of the box is only 30 L, 5 L of the 35 L will not fit into the box.

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Example Question #31 : Cubes

Kim from Idaho can only stack bales of hay in her barn for 3 hours before she needs a break. She stacks the bales at a rate of 2 bales per minute, 3 bales high with 5 bales in a single row. How many full rows will she have at the end of her stacking?

Possible Answers:

Correct answer:

Explanation:

She will stack 360 bales in 3 hours. One row requires 15 bales. 360 divided by 15 is 24.

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

A cube has a volume of $\dpi{100} \small 8 cm^{3}$ . What is the volume of cube with sides that are twice as long?

Possible Answers:

$\dpi{100} \small 64 cm^{3}$

$\dpi{100} \small 12 cm^{3}$

$\dpi{100} \small 27 cm^{3}$

$\dpi{100} \small 16 cm^{3}$

$\dpi{100} \small 2 cm^{3}$

Correct answer:

$\dpi{100} \small 64 cm^{3}$

Explanation:

The volume of a cube is $\dpi{100} \small s^{3}$ .

If each side of the cube is $\dpi{100} \small 2cm$ , then the volume will be $\dpi{100} \small 8cm^{3}$ .

If we double each side, then each side would be $\dpi{100} \small 4cm$ and the volume would be $\dpi{100} \small 64cm^{3}$ .

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Example Question #1381 : Concepts

How many smaller boxes with a dimensions of 1 by 5 by 5 can fit into cube shaped box with a surface area of 150?

Possible Answers:

Correct answer:

Explanation:

There surface are of a cube is 6 times the area of one face of the cube , therefore $6a^{2}=150$

$a^{2}=25$

$a=5$

a is equal to an edge of the cube

the volume of the cube is $a^{3}=5^{3}=125$

The problem states that the dimensions of the smaller boxes are 1 x 5 x 5, the volume of one of the smaller boxes is 25.

Therefore, 125/25 = 5 small boxes

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

If a cube has its edges increased by a factor of 5, what is the ratio of the new volume to the old volume?

Possible Answers:

15:1

100:1

5:1

125:1

25:1

Correct answer:

125:1

Explanation:

A cubic volume is side*side*side . Let the original sides be 1, so that the original volume is 1. Then find the volume if the sides measure 5. This new volume is 125. Therefore, the ratio of new volume to old volume is 125: 1.

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Example Question #731 : Sat Mathematics

A cube is inscribed inside a sphere of radius 1 such that each of the eight vertices of the cube lie on the surface of the sphere. What is the volume of the cube?

Possible Answers:

$\frac{10\sqrt{3}}{7}$

$\frac{8\sqrt{3}}{9}$

$\frac{7\sqrt{3}}{10}$

$\frac{9\sqrt{3}}{8}$

Correct answer:

$\frac{8\sqrt{3}}{9}$

Explanation:

Cube

To make this problem easier to solve, we can inscribe a smaller square in the cube. In the diagram above, points A, B, C are midpoints of the edges of the inscribed cube. Therefore point , a vertex of the smaller cube, is also the center of the sphere. Point lies on the circumference of the sphere, so ED = 1 . is also the hypotenuse of right triangle FED . Similarly, is the hypotenuse of right triangle FBD . If we let BD = x , then, by the properties of a right triangle, $FD = x\sqrt2$ .

Using the Pythagorean Theorem, we can now solve for :

FD^2 + FE^2 = DE^2

$(x\sqrt{2})^2 + x^2 = 1^2$

2x^2 + x^2 = 1

3x^2 = 1

$x^2 = \frac{1}{3}$

$x=\sqrt{\frac{1}{3}}$

Since the side of the inscribed cube is , the volume is $V= (2x)^3=8x^3=8\sqrt{\frac{1}{3}}^3=\frac{8}{3\sqrt{3}}=\frac{8}{3\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{8\sqrt{3}}{9}$ .

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Example Question #1391 : Concepts

A perfect cube has a volume of 8 cubic centimeters. If the height, length and width of the cube were doubled, what would be the volume of the cube?

Possible Answers:

$48\:cm^3$

$128\:cm^3$

$16\:cm^3$

$64\:cm^3$

$24\:cm^3$

Correct answer:

$64\:cm^3$

Explanation:

Volume is calculated by height x width x length: $V=h\cdot w \cdot l$

For a cube, the height, width, and length are all the same value, so the equation can be simplified to V=s^3 , where is the length of one edge of the cube.

We know that for the initial cube, $V=8\:cm^3$ , so we can substitute this into the volume equation and solve for the length of one of the cube's sides:

V=s^3

8=s^3

$\sqrt[3]8=\sqrt[3]{s^3}$

2=s

So, one edge of the initial cube is $2\:cm$ long. When doubled, the cube will have edges that are each $4\:cm$ long. We can solve for the final volume of the cube by substituting $4\:cm$ into the equation for the volume of a cube and solving:

V=s^3

V=4^3

$V=4\cdot 4\cdot 4=64\:cm^3$

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Example Question #32 : Cubes

What's the volume of a cube with a length of 10a ?

Possible Answers:

30a^3

1000a^3

3600a^2

100a^3

300a^3

Correct answer:

1000a^3

Explanation:

Write the volume formula for a cube.

V=s^3

Substitute the side.

V=s^3=(10a)^3 = 1000a^3

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Example Question #32 : Cubes

If the surface area of a cube is 30, what is the volume of the cube?

Possible Answers:

$5\sqrt5$

$\sqrt5$

$5\sqrt3$

125

900

Correct answer:

$5\sqrt5$

Explanation:

Write the surface area formula of the cube.

A=6s^2

Substitute the surface area and find the side.

30=6s^2

s^2=5

$s=\sqrt5$

Write the volume for the cube.

V=s^3

Substitute the side to the volume formula.

$V=s^3=5\sqrt5$

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SAT Math : Solid Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

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

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

Example Question #31 : Cubes

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

Example Question #1381 : Concepts

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

Example Question #731 : Sat Mathematics

Example Question #1391 : Concepts

Example Question #32 : Cubes

Example Question #32 : Cubes

All SAT Math Resources

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