Final answer:
To find the surface area of the cone, calculate the radius using the slant height and the lateral area. Then, use the formula for surface area to find the final answer.
Step-by-step explanation:
To find the surface area of the cone, we need to first find the radius of the cone. The slant height of the cone can be used to find the radius using the Pythagorean theorem. Let's denote the slant height as s and the radius as r. We have the equation:
s² = r² + h²
Given that the slant height s is 51 cm and the lateral area is 12247 square centimeters, we can use the formula for the lateral area of a cone:
Lateral Area = πrl
Plugging in the values, we have:
12247 = πr(51)
Solving for r, we find the radius of the cone. Once we have the radius, we can calculate the surface area of the cone using the formula:
Surface Area = Lateral Area + πr²
Substituting the values, we can find the surface area of the cone.