Question

A cube and a square-based pyramid have identical heights and the same base area. If the cube's volume is 450 cubic feet, what is the volume of the pyramid?

(1 point)
Responses

153 ft3153 ft3

220 ft3220 ft3

150 ft3150 ft3

50 ft3

Answer 1

To solve the problem, we first need to find the side length of the cube using its volume.

The volume \( V \) of a cube is given by the formula:
\[
V = s^3,
\]
where \( s \) is the length of a side of the cube.

Given that the volume of the cube is 450 cubic feet, we have:
\[
s^3 = 450.
\]
To find \( s \), we calculate:
\[
s = \sqrt[3]{450} \approx 7.6 \text{ feet}.
\]

Now, the base area \( A \) of the cube (which is also the area of its base) is:
\[
A = s^2 = (7.6)^2 \approx 57.76 \text{ square feet}.
\]

Since the cube and the square-based pyramid have the same base area and height, we can also calculate the volume of the square-based pyramid. The volume \( V \) of a pyramid is given by:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}.
\]

The height of the cube is also \( s \), which is approximately \( 7.6 \) feet. Therefore, the volume of the pyramid becomes:
\[
V = \frac{1}{3} \times A \times s,
\]
where \( A = 57.76 \) square feet, and \( s \approx 7.6 \) feet.

Now substitute values:
\[
V = \frac{1}{3} \times 57.76 \times 7.6.
\]
Calculating this:
\[
V = \frac{1}{3} \times 438.176 \approx 146.059 \text{ cubic feet}.
\]

Thus, rounding it leads us to a volume of approximately \( 150 \) cubic feet.

The volume of the pyramid is **150 ft³**. Thus, the correct response is:
**150 ft³**.