Question
rectangular prism has a volume of 252 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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Answers
GPT-4o mini
The volume \( V \) of a rectangular prism can be calculated using the formula:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
For a rectangular pyramid, the volume is given by the formula:
\[
V = \frac{1}{3} \times \text{base area} \times \text{height}
\]
If the pyramid has a base and height that are congruent to the prism, its base area is the same as the base area of the prism.
Let \( B \) denote the base area of the prism (which is the same for the pyramid), and let \( h \) denote the height of both the prism and the pyramid. The volume of the prism is given as:
\[
V_{prism} = B \times h = 252 \, \text{in}^3
\]
The volume of the pyramid can then be calculated as:
\[
V_{pyramid} = \frac{1}{3} \times B \times h
\]
Substituting \( B \times h \) with 252 in the pyramid's volume formula gives:
\[
V_{pyramid} = \frac{1}{3} \times 252 = 84 \, \text{in}^3
\]
So, the volume of the pyramid is:
\[
\boxed{84} \, \text{in}^3
\]
\[
V = \text{length} \times \text{width} \times \text{height}
\]
For a rectangular pyramid, the volume is given by the formula:
\[
V = \frac{1}{3} \times \text{base area} \times \text{height}
\]
If the pyramid has a base and height that are congruent to the prism, its base area is the same as the base area of the prism.
Let \( B \) denote the base area of the prism (which is the same for the pyramid), and let \( h \) denote the height of both the prism and the pyramid. The volume of the prism is given as:
\[
V_{prism} = B \times h = 252 \, \text{in}^3
\]
The volume of the pyramid can then be calculated as:
\[
V_{pyramid} = \frac{1}{3} \times B \times h
\]
Substituting \( B \times h \) with 252 in the pyramid's volume formula gives:
\[
V_{pyramid} = \frac{1}{3} \times 252 = 84 \, \text{in}^3
\]
So, the volume of the pyramid is:
\[
\boxed{84} \, \text{in}^3
\]