To find the surface area of the cone peak, we need to use the formula for the lateral surface area of a cone and add the area of the base:
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Lateral Surface Area of the Cone:
\[ \text{Lateral Surface Area} = \pi r l \]
where \( r \) is the radius and \( l \) is the slant height.
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Area of the Base of the Cone:
\[ \text{Base Area} = \pi r^2 \]
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Total Surface Area:
\[ \text{Total Surface Area} = \text{Lateral Surface Area} + \text{Base Area} \]
Now, we need to calculate the radius \( r \) since we are given the diameter. The diameter is 14 inches, so the radius is:
\[ r = \frac{d}{2} = \frac{14}{2} = 7 \text{ inches} \]
Given:
- \( r = 7 \) inches
- \( l = 20 \) inches
- \( \pi \approx 3.14 \)
Now, we will compute the lateral surface area first:
\[ \text{Lateral Surface Area} = \pi r l = 3.14 \cdot 7 \cdot 20 \]
Calculating that:
\[ \text{Lateral Surface Area} = 3.14 \cdot 7 = 21.98 \]
\[ 21.98 \cdot 20 = 439.6 \text{ square inches} \]
Next, we find the area of the base:
\[ \text{Base Area} = \pi r^2 = 3.14 \cdot (7^2) = 3.14 \cdot 49 \]
Calculating that:
\[ \text{Base Area} = 3.14 \cdot 49 = 153.86 \text{ square inches} \]
Finally, we add the lateral surface area and the base area to find the total surface area of the cone:
\[ \text{Total Surface Area} = \text{Lateral Surface Area} + \text{Base Area} \]
Calculating that:
\[ \text{Total Surface Area} = 439.6 + 153.86 = 593.46 \text{ square inches} \]
Therefore, the surface area of the cone peak is:
\[ \boxed{593.46} \text{ square inches} \]