Question
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)
_
_
Answers
GPT-4o mini
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 \( \boxed{9} \, \text{in}^3 \).
\[
V = \text{base area} \times \text{height}
\]
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 \( \boxed{9} \, \text{in}^3 \).