To find the volume of a cone, we can use the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
Given:
- Radius \( r = 13 \) inches
- Height \( h = 27 \) inches
We can substitute these values into the formula:
\[ V = \frac{1}{3} \pi (13^2)(27) \]
Calculating \( 13^2 \):
\[ 13^2 = 169 \]
Now substitute this value into the volume formula:
\[ V = \frac{1}{3} \pi (169)(27) \]
Now multiply \( 169 \) by \( 27 \):
\[ 169 \times 27 = 4563 \]
Now substitute this back into the volume formula:
\[ V = \frac{1}{3} \pi (4563) \]
Calculating \( \frac{1}{3} \times 4563 \):
\[ \frac{4563}{3} = 1521 \]
Now include \(\pi\):
\[ V = 1521\pi \]
Using the approximate value of \(\pi \approx 3.14159\):
\[ V \approx 1521 \times 3.14159 \approx 4778.36 \text{ cubic inches} \]
Thus, the volume of the cone is approximately 4778.36 cubic inches, which corresponds to the first option.