Question
A right 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?
Answers
GPT-4o mini
The volume \( V \) of a right rectangular prism is calculated using the formula:
\[
V = l \times w \times h
\]
where \( l \), \( w \), and \( h \) are the length, width, and height, respectively. For this problem, we know:
\[
V = 27 \text{ in}^3
\]
Now, the volume \( V \) of a rectangular pyramid is given by 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 congruent to the base of the rectangular prism. Therefore, the area of the base of the pyramid \( B \) is equal to the area of the base of the prism, which we can express mathematically as:
\[
B = l \times w
\]
Given that the height of the pyramid is also equal to the height of the prism (denoted as \( h \)), we can substitute these values into the pyramid's volume formula as follows:
\[
V_{\text{pyramid}} = \frac{1}{3} \times (l \times w) \times h
\]
Next, substituting \( l \times w \times h = 27 \text{ in}^3 \), we can rewrite the volume:
\[
V_{\text{pyramid}} = \frac{1}{3} \times V_{\text{prism}} = \frac{1}{3} \times 27 \text{ in}^3
\]
Calculating this gives:
\[
V_{\text{pyramid}} = \frac{27}{3} \text{ in}^3 = 9 \text{ in}^3
\]
Thus, the volume of the pyramid is:
\[
\boxed{9 \text{ in}^3}
\]
\[
V = l \times w \times h
\]
where \( l \), \( w \), and \( h \) are the length, width, and height, respectively. For this problem, we know:
\[
V = 27 \text{ in}^3
\]
Now, the volume \( V \) of a rectangular pyramid is given by 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 congruent to the base of the rectangular prism. Therefore, the area of the base of the pyramid \( B \) is equal to the area of the base of the prism, which we can express mathematically as:
\[
B = l \times w
\]
Given that the height of the pyramid is also equal to the height of the prism (denoted as \( h \)), we can substitute these values into the pyramid's volume formula as follows:
\[
V_{\text{pyramid}} = \frac{1}{3} \times (l \times w) \times h
\]
Next, substituting \( l \times w \times h = 27 \text{ in}^3 \), we can rewrite the volume:
\[
V_{\text{pyramid}} = \frac{1}{3} \times V_{\text{prism}} = \frac{1}{3} \times 27 \text{ in}^3
\]
Calculating this gives:
\[
V_{\text{pyramid}} = \frac{27}{3} \text{ in}^3 = 9 \text{ in}^3
\]
Thus, the volume of the pyramid is:
\[
\boxed{9 \text{ in}^3}
\]