To find the height of the rectangular prism, we can use the formula for the volume of a rectangular prism, which is:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
We are given the base area of the rectangular prism as \( 15 , \text{cm}^2 \), and we need to find the height \( h \).
Given that the volume of the rectangular prism is equal to the volume of the pyramid plus the volume of the rectangular prism (which is not specified, but we can assume is not part of the question), we only need to find the height from the given base area.
The pyramid with the same base area of \( 15 , \text{cm}^2 \) has a volume of \( 20 , \text{cm}^3 \). The formula for the volume of a pyramid is:
\[ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Let’s denote the height of the pyramid as \( h_p \). Thus:
\[ 20 = \frac{1}{3} \times 15 \times h_p \] \[ 20 = 5 \times h_p \] \[ h_p = \frac{20}{5} = 4 , \text{cm} \]
Since we have the volume of the pyramid, we can now focus on finding the height of the prism. The volume of the prism is typically set in comparison if needed to solve, but the important information is the base area and the lack of additional data on the volume of the prism.
Now solving for height when knowing that the base height corresponds to a variable context, we can use the base area:
\[ \text{Volume of the Prism} = 15 \times h \]
Since we need to find height independently provided pyramid height relates but not to volume addition as per prism defaults.
Given all provided:
Assume that if additional context might suggest to find h, if stand alone would require the device of confirming no additional volumes.
Answer
The simplest presumption based, taking the height of
The height is \( \boxed{4} , \text{cm} \).
None should replace or compare to other unless tied to necessary contextual limits in volume or structural metrics.