To find the height of the rectangular pyramid, we can start by using the information given about the rectangular prism and its relationship with the pyramid.
- Base Area of the Prism: The base area of the rectangular prism is given as \(25 , \text{in}^2\).
- Volume of the Prism: The volume of the rectangular prism is given as \(125 , \text{in}^3\).
The formula for the volume of a rectangular prism is given by:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
Substituting the known values into the formula, we get:
\[ 125 , \text{in}^3 = 25 , \text{in}^2 \times \text{Height} \]
To solve for the height of the prism, we can rearrange the formula:
\[ \text{Height} = \frac{125 , \text{in}^3}{25 , \text{in}^2} = 5 , \text{in} \]
So, the height of the rectangular prism is \(5 , \text{in}\).
- Volume of the Pyramid: For the rectangular pyramid, which has a congruent base and a height equal to that of the prism, the volume is calculated using the formula:
\[ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
We already know:
- Base Area = \(25 , \text{in}^2\)
- Height = \(5 , \text{in}\)
Now substituting these values into the pyramid volume formula:
\[ \text{Volume} = \frac{1}{3} \times 25 , \text{in}^2 \times 5 , \text{in} \]
Calculating this gives:
\[ \text{Volume} = \frac{1}{3} \times 125 , \text{in}^3 = \frac{125}{3} , \text{in}^3 \]
Thus, the height of the pyramid, considering the fact that it matches the height of the prism, is:
\[ \text{Height of the pyramid} = 5 , \text{in} \]
So the height of the pyramid is also \(5 , \text{in}\).