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)

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1 answer

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

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

Given the volume of the rectangular prism is \( 27 , \text{in}^3 \), let's denote the base area of the prism as \( B \) and the height as \( h \):

\[ B \times h = 27 \]

The volume \( V \) of a rectangular pyramid is calculated using the formula:

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

Since the base area and height of the pyramid are congruent to those of the prism, we can use the same base area \( B \) and height \( h \) for the pyramid. Therefore, the volume of the rectangular pyramid is:

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

Substituting the expression for \( B \times h \) from the prism's volume:

\[ V = \frac{1}{3} \times 27 \]

Calculating this gives:

\[ V = 9 , \text{in}^3 \]

Thus, the volume of the pyramid is \( \boxed{9} , \text{in}^3 \).