To find the height of the rectangular prism, we can use the formula for the volume of a rectangular prism:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
Given that the base area of the rectangular prism is \(15 , \text{cm}^2\) and we need to find the height, we can express the volume of the rectangular prism as follows:
\[ \text{Volume of the rectangular prism} = 15 , \text{cm}^2 \times \text{Height} \]
We also have the volume of the pyramid, which is given by the formula:
\[ \text{Volume of the pyramid} = \frac{1}{3} \times \text{Base Area} \times \text{Height of the pyramid} \]
We know that the base area is congruent to that of the rectangular prism and the volume of the pyramid is \(20 , \text{cm}^3\). Therefore:
\[ 20 , \text{cm}^3 = \frac{1}{3} \times 15 , \text{cm}^2 \times \text{Height of the pyramid} \]
Now, solving for the height of the pyramid:
\[ 20 = \frac{1}{3} \times 15 \times \text{Height of the pyramid} \] \[ 20 = 5 \times \text{Height of the pyramid} \] \[ \text{Height of the pyramid} = \frac{20}{5} = 4 , \text{cm} \]
Since the height of the pyramid and the rectangular prism is said to be congruent, the height of the rectangular prism is also \(4 , \text{cm}\).
Thus, the height of the rectangular prism is \( \boxed{4 , \text{cm}} \).