To find the height of the rectangular prism, we need to first understand the relationship between the base area and the volume.
The volume \( V \) of a rectangular prism is given by the formula:
\[ V = \text{Base Area} \times \text{Height} \]
Given that the base area of the prism is \( 15 , \text{cm}^2 \) and we are not directly given its volume, we need to analyze the information we have about the pyramid.
The volume of a pyramid is given by the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
From the problem, we know that this pyramid has a volume of \( 20 , \text{cm}^3 \) and has a base area congruent to that of the rectangular prism, which is also \( 15 , \text{cm}^2 \).
Plugging into the pyramid volume formula, we have:
\[ 20 = \frac{1}{3} \times 15 \times h \]
Where \( h \) is the height of the pyramid. To solve for \( h \):
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Rearranging gives: \[ 20 = 5h \]
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Now, solve for \( h \): \[ h = \frac{20}{5} = 4 , \text{cm} \]
Since the problem is asking for the height of the rectangular prism and does not state any difference between the dimensions for the two shapes aside from their definitions, we can reasonably conclude that the height of the rectangular prism can also be taken as \( 4 , \text{cm} \).
Therefore, the height of the rectangular prism is:
A) 4 cm