A rectangular prism has a volume of 27 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

To find the volume of the rectangular pyramid with a base and height congruent to the rectangular prism, we can use the formula for the volume of a rectangular prism and the formula for the volume of a pyramid.

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

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

For the rectangular pyramid, the volume \( V_p \) is given by:

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

Since the base of the pyramid is congruent to the base of the prism, the base area can be expressed as:

\[ \text{base area} = \text{length} \times \text{width} \]

Thus, the volume of the pyramid can be written as:

\[ V_p = \frac{1}{3} \times (\text{length} \times \text{width}) \times \text{height} \]

Now, we know that:

\[ \text{length} \times \text{width} \times \text{height} = 27 , \text{in}^3 \]

This means:

\[ \text{length} \times \text{width} = \frac{27}{\text{height}} \]

Now substituting this into the pyramid volume formula:

\[ V_p = \frac{1}{3} \times \left(\frac{27}{\text{height}}\right) \times \text{height} \]

This simplifies to:

\[ V_p = \frac{1}{3} \times 27 = 9 , \text{in}^3 \]

Thus, the volume of the pyramid is:

\[ \boxed{9 , \text{in}^3} \]