Question

What is the volume of a rectangular prism with a congruent base and height to a rectangular pyramid with a volume of 10 cm^3?

Answer 1

To find the volume of the rectangular prism that has a congruent base and height to a rectangular pyramid with a volume of 10 cm³, we first need to establish the relationship between the volumes of the pyramid and the prism.

The volume \( V \) of a rectangular pyramid can be expressed as:

\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]

Given that the volume of the pyramid is 10 cm³:

\[
10 = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]

From this, we can derive:

\[
\text{Base Area} \times \text{Height} = 30 \quad \text{(multiplying both sides by 3)}
\]

Next, the volume \( V \) of a rectangular prism is given by:

\[
V = \text{Base Area} \times \text{Height}
\]

Since the prism has the same base area and height as the pyramid, we can substitute \( \text{Base Area} \times \text{Height} \) derived from the pyramid's volume into the prism's formula:

\[
V = \text{Base Area} \times \text{Height} = 30 \, \text{cm}^3
\]

Thus, the volume of the rectangular prism is:

\[
\boxed{30 \, \text{cm}^3}
\]