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?

Answer 1

We are given that the volume of the cube is 450 cubic feet, and we need to find the volume of a square-based pyramid that has the same base area and height as the cube.

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

Given that the volume of the cube is 450 cubic feet, we can write:
\[
s^3 = 450
\]
To find the side length \( s \), we take the cube root:
\[
s = \sqrt[3]{450}
\]

2. **Base Area of the Cube**:
The base area \( A \) of the cube is:
\[
A = s^2
\]
So substituting \( s = \sqrt[3]{450} \):
\[
A = \left(\sqrt[3]{450}\right)^2 = \sqrt[3]{450^2}
\]

3. **Height of the Cube**:
The height of the cube is simply equal to its side length \( s \):
\[
h = s = \sqrt[3]{450}
\]

4. **Volume of the Pyramid**:
The volume \( V_p \) of a square-based pyramid is given by:
\[
V_p = \frac{1}{3} A h
\]
where \( A \) is the base area and \( h \) is the height.

Using the base area \( A = s^2 \) and height \( h = s \), we substitute these into the volume formula:
\[
V_p = \frac{1}{3} (s^2)(s) = \frac{1}{3} s^3
\]

Since \( s^3 = 450 \), we have:
\[
V_p = \frac{1}{3} \times 450 = 150
\]

Thus, the volume of the pyramid is \(\boxed{150}\) cubic feet.