To find the height of the rectangular prism, we can use the formula for the volume of a rectangular prism:
\[ V = \text{Base Area} \times \text{Height} \]
We know from the problem that the base area of the rectangular prism is \( 15 , \text{cm}^2 \).
Let \( h \) be the height of the rectangular prism. Thus, the volume of the rectangular prism can be expressed as:
\[ V = 15 , \text{cm}^2 \times h \]
Now we also have information about the pyramid. The volume of a pyramid is given by the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
For the pyramid, the base area is also \( 15 , \text{cm}^2 \), and its volume is given as \( 20 , \text{cm}^3 \). Therefore, we can set up the equation for the pyramid:
\[ 20 , \text{cm}^3 = \frac{1}{3} \times 15 , \text{cm}^2 \times h_p \]
Where \( h_p \) is the height of the pyramid. We can solve for \( h_p \):
\[ 20 = \frac{15}{3} \times h_p \]
\[ 20 = 5 \times h_p \]
\[ h_p = \frac{20}{5} = 4 , \text{cm} \]
Now that we have the height of the pyramid \( h_p = 4 , \text{cm} \), we can return to the volume of the rectangular prism. Assuming it shares the same height, we solve for \( h \):
\[ V = 15 , \text{cm}^2 \times h \]
We don't have the volume of the rectangular prism, but if we assume the height must relate to the pyramid's while using the given base area, we denote its height directly depending on context or any additional provided parameters.
If we needed a definitive answer, we might require either the volume for the rect prism or additional context from the original problem. For now, let's assert height \( h \):
Assuming equal height, \( h = 4 , \text{cm} \).
So the height of the rectangular prism can be:
\[ \text{Height of the rectangular prism} = 4 , \text{cm} (assuming equality with pyramid height for context-clearness). \]