rectangular prism has a volume of 252 in.3 . If a rectangular pyramid has a base and height congruent to the prism, what is the volume of the pyramid?(1 point)

___________

1 answer

The volume \( V \) of a rectangular prism can be calculated using the formula:

\[ V = \text{length} \times \text{width} \times \text{height} \]

For a rectangular pyramid, the volume is given by the formula:

\[ V = \frac{1}{3} \times \text{base area} \times \text{height} \]

If the pyramid has a base and height that are congruent to the prism, its base area is the same as the base area of the prism.

Let \( B \) denote the base area of the prism (which is the same for the pyramid), and let \( h \) denote the height of both the prism and the pyramid. The volume of the prism is given as:

\[ V_{prism} = B \times h = 252 , \text{in}^3 \]

The volume of the pyramid can then be calculated as:

\[ V_{pyramid} = \frac{1}{3} \times B \times h \]

Substituting \( B \times h \) with 252 in the pyramid's volume formula gives:

\[ V_{pyramid} = \frac{1}{3} \times 252 = 84 , \text{in}^3 \]

So, the volume of the pyramid is:

\[ \boxed{84} , \text{in}^3 \]