To find the height of the rectangular pyramid, we start with the information given about the rectangular prism:
- Base area of the prism \( A = 25 , \text{in}^2 \)
- Volume of the prism \( V = 125 , \text{in}^3 \)
The formula for the volume of a rectangular prism is given by:
\[ V = A \times h \]
where \( A \) is the base area and \( h \) is the height of the prism. We can use this formula to find the height \( h \) of the prism:
\[ 125 = 25 \times h \]
Solving for \( h \):
\[ h = \frac{125}{25} = 5 , \text{in} \]
Now, we need to find the height of the rectangular pyramid, which has the same base area and height as the prism. The volume \( V \) of a pyramid is given by the formula:
\[ V = \frac{1}{3} A \times h \]
Since the base area and height of the pyramid are congruent to those of the prism, we will use \( A = 25 , \text{in}^2 \) and \( h = 5 , \text{in} \). However, we need to find the height of the pyramid; we can denote this height as \( h_p \). The volume of the pyramid can also be expressed as:
\[ V = \frac{1}{3} \times 25 \times h_p \]
Since the height of the pyramid is still 5 inches (the height is congruent), we substitute \( h_p = 5 \):
Calculating the volume of the pyramid using the base area:
\[ V = \frac{1}{3} \times 25 \times 5 \]
Calculating:
\[ V = \frac{1}{3} \times 125 = \frac{125}{3} , \text{in}^3 \]
Thus, the height of the rectangular pyramid is also congruent to the height of the rectangular prism, which is:
\[ \text{Height of the pyramid} = 5 , \text{in} \]
Therefore, the height of the pyramid is \( \boxed{5} \) inches.