The volume \( V \) of a pyramid varies jointly with its base area \( B \) and its height \( h \), which can be expressed mathematically as:
\[ V = k \cdot B \cdot h \]
where \( k \) is a constant.
From the information given, we know that when the base area \( B = 24 \) square inches and the height \( h = 3 \) inches, the volume \( V = 24 \) cubic inches.
Let's first find the constant \( k \):
\[ 24 = k \cdot 24 \cdot 3 \]
This simplifies to:
\[ 24 = k \cdot 72 \]
Solving for \( k \):
\[ k = \frac{24}{72} = \frac{1}{3} \]
Now that we have \( k \), we can use this value to find the volume of a pyramid with a base area of \( 15 \) square inches and a height of \( 7 \) inches.
Substitute \( B = 15 \) and \( h = 7 \) into the volume formula:
\[ V = \frac{1}{3} \cdot 15 \cdot 7 \]
Calculating this:
\[ V = \frac{1}{3} \cdot 105 = 35 \]
Therefore, the volume of the pyramid is:
\[ \boxed{35} \text{ cubic inches.} \]