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 of the rectangular prism is \(15 , cm^2\), and we need to find the height \(h\).
Let's denote the volume of the rectangular prism as \(V_p\). To find \(h\), we can rearrange the formula:
\[ h = \frac{V_p}{\text{Base Area}} \]
Next, we know that the pyramid with a congruent base has a volume of \(20 , cm^3\). The volume of a pyramid is given by the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Substituting the known values for the pyramid:
\[ 20 = \frac{1}{3} \times 15 \times h_p \]
Where \(h_p\) is the height of the pyramid. We can solve for \(h_p\):
\[ 20 = 5h_p \] \[ h_p = \frac{20}{5} = 4 , cm \]
Since the pyramid's height is 4 cm and the prism's height should be calculated separately, we need to find the volume or use the value:
To find the height of the rectangular prism, we can set its volume equal to \(15 \times h\):
We can rearrange the volume for the prism itself by assuming a reasonable guess or comparing volumes; however, since no volume was provided for the prism (and assuming it doesn't relate directly to \(h_p\) based on base area), we can continue:
Let's denote the height of the rectangular prism as \(h\).
Given the base area of the rectangular prism is \(15 , cm^2\), substituting the formula:
Now let's set a practical volume estimate or height from the response options against the known height expressions:
Trying a calculated base observation may indicate based on those specific number responses directly suggests across contextual checks. The Base responses indicate height features might outright be linked easily to the height of \(4 , cm\).
From the list provided;
The height of the rectangular prism is:
4 cm.