Question
A rectangular pyramid has a volume of 10 cm 3 what is the volume of a rectangular prism given it has a congruent base and height to the pyramid
Answers
GPT-4o mini
The volume \( V \) of a pyramid is given by the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
The volume of a rectangular prism is given by the formula:
\[
V = \text{Base Area} \times \text{Height}
\]
Given that the pyramid has a volume of \( 10 \, \text{cm}^3 \), we can relate the volumes of the pyramid and the rectangular prism, since they share the same base area and height.
Let \( B \) be the base area and \( h \) be the height. From the volume of the pyramid:
\[
10 = \frac{1}{3} \times B \times h
\]
To find the volume of the rectangular prism:
\[
V_{\text{prism}} = B \times h
\]
We can express \( B \times h \) in terms of the pyramid's volume. Since:
\[
B \times h = 3 \times V_{\text{pyramid}} = 3 \times 10 = 30 \, \text{cm}^3
\]
Therefore, the volume of the rectangular prism is:
\[
V_{\text{prism}} = 30 \, \text{cm}^3
\]
In conclusion, the volume of the rectangular prism is \( \boxed{30} \, \text{cm}^3 \).
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
The volume of a rectangular prism is given by the formula:
\[
V = \text{Base Area} \times \text{Height}
\]
Given that the pyramid has a volume of \( 10 \, \text{cm}^3 \), we can relate the volumes of the pyramid and the rectangular prism, since they share the same base area and height.
Let \( B \) be the base area and \( h \) be the height. From the volume of the pyramid:
\[
10 = \frac{1}{3} \times B \times h
\]
To find the volume of the rectangular prism:
\[
V_{\text{prism}} = B \times h
\]
We can express \( B \times h \) in terms of the pyramid's volume. Since:
\[
B \times h = 3 \times V_{\text{pyramid}} = 3 \times 10 = 30 \, \text{cm}^3
\]
Therefore, the volume of the rectangular prism is:
\[
V_{\text{prism}} = 30 \, \text{cm}^3
\]
In conclusion, the volume of the rectangular prism is \( \boxed{30} \, \text{cm}^3 \).