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 that the base area of the rectangular prism is \(15 , \text{cm}^2\). Let's denote the height of the rectangular prism as \(h\). Therefore, the volume of the rectangular prism can be expressed as:
\[ \text{Volume of rectangular prism} = 15 , \text{cm}^2 \times h \]
Next, we need the height \(h\). However, we do not directly know the volume of the rectangular prism. The problem gives us the volume of a pyramid which has a congruent base and height. The volume of a pyramid is given by the formula:
\[ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
For the pyramid provided, we know the volume is \(20 , \text{cm}^3\) and the base area is also \(15 , \text{cm}^2\). Let's denote the height of the pyramid also as \(h_p\).
Using the volume formula for the pyramid:
\[ 20 , \text{cm}^3 = \frac{1}{3} \times 15 , \text{cm}^2 \times h_p \]
Now, we can solve for \(h_p\):
\[ 20 = \frac{15}{3} \times h_p \] \[ 20 = 5 \times h_p \] \[ h_p = \frac{20}{5} = 4 , \text{cm} \]
Since the pyramid has a height of \(4 , \text{cm}\) and a congruent base to that of the rectangular prism, it suggests that this height can be used directly. However, we still need to find the height of the rectangular prism itself.
Using the height of the prism \(h\):
\[ \text{Volume of rectangular prism} = 15 \times h \]
To find its height \(h\), we don't know the volume of the rectangular prism. However, let us assume typical dimensions. Without the volume, we rely on that the prism must have dimensions such that a logical value for \(h\) can be selected.
Thus, if we analyze through given choices, let's check back solutions against plausible height choices according to respective metrics.
The answer choices are:
- 4 cm
- 5 cm
- \( \frac{4}{3} \) cm
- 20 cm
Among these, knowing volumes and corresponding practical measures weight against reasonable common prism heights, we can choose:
\(\text{Height of the rectangular prism would be } 4 , \text{cm}\) which fits as the ideal height.
Thus, the height of the rectangular prism is 4 cm.