How to find the volume of a cube - PSAT Math

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Question

A cubic box has sides of length x. Another cubic box has sides of length 4_x_. How many of the boxes with length x could fit in one of the larger boxes with side length 4_x_?

Answer

The volume of a cubic box is given by (side length)3. Thus, the volume of the larger box is (4_x_)3 = 64_x_3, while the volume of the smaller box is _x_3. Divide the volume of the larger box by that of the smaller box, (64_x_3)/(_x_3) = 64.

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Question

I have a hollow cube with 3” sides suspended inside a larger cube of 9” sides. If I fill the larger cube with water and the hollow cube remains empty yet suspended inside, what volume of water was used to fill the larger cube?

Answer

Determine the volume of both cubes and then subtract the smaller from the larger. The large cube volume is 9” * 9” * 9” = 729 in3 and the small cube is 3” * 3” * 3” = 27 in3. The difference is 702 in3.

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Question

A cube weighs 5 pounds. How much will a different cube of the same material weigh if the sides are 3 times as long?

Answer

A cube that has three times as long sides is 3x3x3=27 times bigger than the original. Therefore, the answer is 5x27= 135.

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Question

If the volume of a cube is 50 cubic feet, what is the volume when the sides double in length?

Answer

Using S as the side length in the original cube, the original is sss. Doubling one side and tripling the other gives 2s2s2s for a new volume formula for 8sss, showing that the new volume is 8x greater than the original.

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Question

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?

Answer

To make this problem easier to solve, we can inscribe a smaller square in the cube. In the diagram above, points 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 . is also the hypotenuse of right triangle . Similarly, is the hypotenuse of right triangle . If we let , then, by the properties of a right triangle, $FD = x\sqrt2$ .

Using the Pythagorean Theorem, we can now solve for :

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

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

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

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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Question

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?

Answer

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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Question

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?

Answer

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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Question

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?

Answer

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

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Question

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?

Answer

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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Question

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

Answer

A cubic volume is . 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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Question

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?

Answer

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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Question

If a cube has a surface area of , what is the difference between the volume of the cube and the surface area of the cube?

Answer

If the surface area is $600\textup{ sq in}$ , then the area of one face must be $100\textup{ sq in}$ . Therefore, the length of one edge must be $10\textup{ in}$ This means that the volume of the cube is $1000\textup{ cubic in}$ . We can now solve with:

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