The volume \( V \) of a pyramid can be expressed with the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Since the volume varies jointly with the base area and height, we can also represent it as:
\[ V = k \cdot A \cdot h \]
where \( k \) is a constant of proportionality, \( A \) is the base area, and \( h \) is the height.
From the given information, we know:
- Volume \( V = 25 \) cubic inches
- Base area \( A = 15 \) square inches
- Height \( h = 7 \) inches
We can substitute these values into the formula to find \( k \):
\[ 25 = k \cdot 15 \cdot 7 \]
Calculating \( 15 \cdot 7 \):
\[ 15 \cdot 7 = 105 \]
Now we can substitute this into the equation:
\[ 25 = k \cdot 105 \]
To find \( k \), divide both sides by 105:
\[ k = \frac{25}{105} = \frac{5}{21} \]
Now that we have \( k \), we can find the volume of a new pyramid with a base area of \( 36 \) square inches and a height of \( 5 \) inches. Thus, we will set up the equation:
\[ V = k \cdot A \cdot h = \frac{5}{21} \cdot 36 \cdot 5 \]
Now calculate \( 36 \cdot 5 \):
\[ 36 \cdot 5 = 180 \]
Now substitute that back in:
\[ V = \frac{5}{21} \cdot 180 \]
Now we can compute:
\[ V = \frac{5 \cdot 180}{21} = \frac{900}{21} \]
Now simplify \( \frac{900}{21} \):
\[ 900 \div 21 \approx 42.8571 \text{ (This is approximately, but let's reduce it properly)} \]
Finding the GCD of 900 and 21, which is 3:
\[ 900 = 3 \cdot 300,\quad 21 = 3 \cdot 7 \]
So we can simplify:
\[ \frac{900 \div 3}{21 \div 3} = \frac{300}{7} \]
So the volume is:
\[ \frac{300}{7} \text{ cubic inches or approximately } 42.8571 \text{ cubic inches.} \]
Thus, the volume of the pyramid is
\[ \boxed{\frac{300}{7}} \text{ cubic inches.} \]
As a decimal, this is approximately \( 42.86 \) cubic inches.