The volume \( V \) of a pyramid can be expressed with the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
We know that the volume varies jointly with the base area and height, so we can express this relationship as:
\[ V = k \times (\text{Base Area}) \times (\text{Height}) \]
for some constant \( k \).
From the given information, the volume of one pyramid is 35 cubic inches with a base area of 15 square inches and a height of 7 inches. We can use this information to find \( k \):
\[ 35 = k \times 15 \times 7 \]
Calculating \( 15 \times 7 \):
\[ 15 \times 7 = 105 \]
Now, substituting that back into the equation:
\[ 35 = k \times 105 \]
To find \( k \), we divide both sides by 105:
\[ k = \frac{35}{105} = \frac{1}{3} \]
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. The volume will be:
\[ V = \frac{1}{3} \times 36 \times 5 \]
Calculating \( 36 \times 5 \):
\[ 36 \times 5 = 180 \]
Now substituting back into the volume equation:
\[ V = \frac{1}{3} \times 180 = 60 \]
Thus, the volume of the pyramid is:
\[ \text{The volume of the pyramid is } 60 \text{ cubic inches.} \]
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