To find the surface area of a cone, we need to calculate both the lateral surface area and the area of the base (if required). In this case, we are specifically looking for the lateral surface area since it resembles a peak.
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.
We are given:
- The diameter of the cone is 14 inches, so the radius \( r \) is half of that:
\[ r = \frac{14}{2} = 7 \text{ inches} \]
- The slant height \( l \) is given as 20 inches.
Now we can substitute the values into the formula:
\[ A = \pi r l = 3.14 \times 7 \times 20 \]
Calculating that step-by-step:
- Calculate \( 7 \times 20 \):
\[ 7 \times 20 = 140 \]
- Then multiply by \( \pi \) (using \( 3.14 \)):
\[ A = 3.14 \times 140 = 439.6 \]
Thus, the lateral surface area of the cone peak is:
\[ \boxed{439.60} \text{ square inches} \]
This is the surface area of the cone peak rounded to the nearest hundredth.