To find the height of the rectangular prism, we can use the formula for the volume of a rectangular prism, which is given by:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
We know the base area of the rectangular prism is \(15 , \text{cm}^2\). Let \(h\) be the height of the rectangular prism. Then the volume \(V\) of the rectangular prism can be expressed as:
\[ V = 15 \times h \]
We don’t have the volume of the rectangular prism, but we do have information about the pyramid. The volume \(V_p\) of a pyramid is given by:
\[ V_p = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
The volume of the pyramid is \(20 , \text{cm}^3\), and it has a congruent base with the same base area as the prism, which is \(15 , \text{cm}^2\). Let \(h_p\) be the height of the pyramid. Using the formula for the volume of the pyramid, we have:
\[ 20 = \frac{1}{3} \times 15 \times h_p \]
Now, solving for \(h_p\):
\[ 20 = 5h_p \]
Dividing both sides by \(5\):
\[ h_p = \frac{20}{5} = 4 , \text{cm} \]
With the height of the pyramid known, we note that the base areas are congruent but the height of the rectangular prism is not directly related to the height of the pyramid. Thus, we just need to find \(h\) as follows:
Using the volume of the rectangular prism, we need to rearrange our initial equation:
\[ h = \frac{V}{\text{Base Area}} = \frac{V}{15} \]
Since we do not have the volume (V) given for the rectangular prism, we cannot uniquely determine a numerical result from the choices provided without additional information; however, since we have \(V = 20\) for the pyramid, we can conclude that the height of the rectangular prism can also be calculated given a different volume value.
Let's consider the answers provided and use trial for volume = 60 cm³, allowing more flexibility since the height could be:
Then if V= 60 cm³: \(h = \frac{60}{15} = 4\).
So, while we need volume specifically, based on the options, the only plausible height of the rectangular prism is:
\[ 4 , \text{cm} \]
So the correct answer would be 4 cm.