To find the height of the rectangular pyramid, let's first recall the formulas for the volume of a prism and a pyramid.
The volume \( V \) of a prism is calculated using the formula: \[ V = \text{Base Area} \times \text{Height} \]
For the rectangular prism, we have: \[ V_{\text{prism}} = 125 , \text{in}^3 \] \[ \text{Base Area} = 25 , \text{in}^2 \] Let \( h_p \) be the height of the prism (which is equal to the height of the pyramid). Therefore, we can set up the equation: \[ 125 = 25 \times h_p \]
To find \( h_p \), we can solve for it: \[ h_p = \frac{125}{25} = 5 , \text{in} \]
Now, we have the height of the pyramid: \[ h_p = 5 , \text{in} \]
Next, we calculate the volume of the pyramid. The volume \( V \) of a pyramid is given by the formula: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Since the base area and height of the pyramid are congruent to those of the prism, we substitute: \[ V_{\text{pyramid}} = \frac{1}{3} \times 25 \times 5 \] \[ V_{\text{pyramid}} = \frac{1}{3} \times 125 = \frac{125}{3} , \text{in}^3 \]
Despite calculating the volume of the pyramid, remember, the question asked for the height of the pyramid, which, as derived, is: \[ \text{Height of the pyramid} = 5 , \text{in} \]
Thus, the final answer is: \[ \boxed{5} \]