To find the height of the rectangular prism, we can use the formula for the volume of a rectangular prism, which is given by:
\[ V = \text{Base Area} \times \text{Height} \]
We know the base area is \( 15 , \text{cm}^2 \). Let \( h \) be the height of the rectangular prism. Therefore, we can write the equation for the volume of the prism as:
\[ V_{\text{prism}} = 15 , \text{cm}^2 \times h \]
The volume of the pyramid is given as \( 20 , \text{cm}^3 \). The volume of a pyramid is calculated using the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Since the base area of the pyramid is congruent to the prism's base area, we have:
\[ V_{\text{pyramid}} = \frac{1}{3} \times 15 , \text{cm}^2 \times h \]
Setting \( V_{\text{pyramid}} \) to \( 20 , \text{cm}^3 \):
\[ 20 = \frac{1}{3} \times 15 \times h \]
Now, multiply both sides by 3:
\[ 60 = 15 \times h \]
Next, divide both sides by 15 to solve for \( h \):
\[ h = \frac{60}{15} = 4 , \text{cm} \]
Thus, the height of the rectangular prism is:
\[ \boxed{4 , \text{cm}} \]