Question

If the volume of a rectangular prism is 600 in(3), what is the volume of a rectangular pyramid whose base and height is congruent to the prism?
200 in
1,200 in
1,800 in
300 in

Answer 1

The volume \( V \) of a rectangular prism is given by the formula:

\[
V = l \times w \times h
\]

where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

For a rectangular pyramid, the volume is calculated using the formula:

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

where \( B \) is the area of the base and \( h \) is the height of the pyramid.

In this case, the base of the pyramid is the same as the base of the prism, and the height is also the same. The base area \( B \) of the prism (and thus the pyramid) can be calculated as \( B = l \times w \).

Given the volume of the prism is 600 in³, thus:

\[
600 = l \times w \times h
\]

When substituting for the pyramid's volume using the same base area \( B \):

\[
V = \frac{1}{3} \times (l \times w) \times h
\]

We can substitute \( B = l \times w \) into the pyramid's volume formula:

\[
V = \frac{1}{3} \times (l \times w) \times h
\]

From the prism, we have:

\[
l \times w \times h = 600
\]

Thus, we rewrite the volume of the pyramid:

\[
V = \frac{1}{3} \times 600 = 200 \text{ in}^3
\]

Therefore, the volume of the rectangular pyramid is:

\[
\boxed{200 \text{ in}^3}
\]