PSAT Math - How to find the volume of a cube

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?

Explanation

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:

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

100 cu ft

200 cu ft

300 cu ft

400 cu ft

500 cu ft

Explanation

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

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?

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

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

10 pounds

15 pounds

45 pounds

135 pounds

Explanation

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

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?

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

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

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

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

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

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

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?

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.

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?

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

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

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

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

Explanation

Cube

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 $V= (2x)^3=8x^3=8\sqrt{\frac{1}{3}}^3=\frac{8}{3\sqrt{3}}=\frac{8}{3\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{8\sqrt{3}}{9}$ .

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?

Explanation

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

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?

30 lites

1 liters

5 liters

15 liters

No water will flow out of the box

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.

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

Explanation

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.

How to find the volume of a cube

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