A cone has a surface area of 24​𝜋 square feet. Its base has a circumference of 6​𝜋 feet. What is the slant height of the cone?

Sure, let's solve this step by step.

Step 1: Find the radius of the base of the cone. The formula for the circumference of a circle is C = 2πr. Given that the circumference is 6π feet, we can set up the equation 2πr = 6π and solve for r.

r = 6π / 2π = 3 feet

Step 2: Find the area of the base of the cone. The formula for the area of a circle is A = πr². Substituting r = 3 feet into the formula, we get:

A = π(3 feet)² = 9π square feet

Step 3: Find the lateral surface area of the cone. The total surface area of a cone is the sum of the base area and the lateral surface area. Given that the total surface area is 24π square feet and the base area is 9π square feet, we can set up the equation 24π = 9π + Lateral Surface Area and solve for the lateral surface area.

Lateral Surface Area = 24π - 9π = 15π square feet

Step 4: Find the slant height of the cone. The formula for the lateral surface area of a cone is Lateral Surface Area = πrs, where r is the radius and s is the slant height. Substituting r = 3 feet and Lateral Surface Area = 15π square feet into the formula, we get:

15π = π(3 feet)s

Solving for s, we get:

s = 15π / (3π) = 5 feet

So, the slant height of the cone is 5 feet.