To find the surface area of the cone, we need to calculate the lateral surface area (which is the area of the cone's side) and add the area of the base (if required). However, since the problem only mentions the peak and does not specify adding a base, we'll calculate only the lateral surface area.
The formula for the lateral surface area \( A \) of a cone is given by:
\[ A = \pi r l \]
where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone.
- Find the radius: The diameter of the cone is 14 inches, so the radius \( r \) is half of the diameter:
\[ r = \frac{d}{2} = \frac{14}{2} = 7 \text{ inches} \]
-
Use the slant height: The slant height \( l \) is given as 20 inches.
-
Calculate the lateral surface area:
\[ A = \pi r l = 3.14 \times 7 \times 20 \]
- Calculate:
\[ A = 3.14 \times 7 = 21.98 \]
\[ A = 21.98 \times 20 = 439.6 \]
Thus, the lateral surface area of the cone peak is approximately:
\[ \boxed{439.60} \text{ square inches} \]
Rounding to the nearest hundredth, the answer remains \( 439.60 \) square inches.