To find the missing slant height \( l \) of the cone, we can use the formula for the surface area of a cone, which includes the base area and the lateral area. The total surface area \( S_A \) of a cone is given by:
\[ S_A = \pi r^2 + \pi r l \]
Where:
- \( S_A \) is the total surface area,
- \( r \) is the radius of the base,
- \( l \) is the slant height.
From the problem, we have:
- \( S_A = 364\pi \) square inches,
- \( r = 13 \) inches.
First, we can find the area of the base:
\[ \text{Area of the base} = \pi r^2 = \pi (13)^2 = \pi \times 169 = 169\pi \text{ square inches.} \]
Now, we can substitute the area of the base into the total surface area formula:
\[ S_A = \pi r^2 + \pi r l \]
Substituting the values we have:
\[ 364\pi = 169\pi + \pi (13)l \]
We can simplify this equation:
\[ 364\pi - 169\pi = \pi (13)l \]
\[ 195\pi = 13\pi l \]
Now, we can divide both sides by \( \pi \):
\[ 195 = 13l \]
Now, solving for \( l \):
\[ l = \frac{195}{13} = 15 \]
Thus, the missing slant height \( l \) is:
\[ \boxed{15} \text{ inches.} \]