Final answer:
The total surface area of a cone with a radius of 3 ft and a slant height of 4 ft is calculated using the formula for the lateral surface area (πrε) plus the area of the base (πr²), which equals 21π ft².
Step-by-step explanation:
To find the total surface area of a cone, we use the provided formula, which consists of two parts: the lateral surface area (the side of the cone) and the area of the base.
The lateral surface area of a cone can be calculated using the formula SA = πrε, where 'r' is the radius of the base and 'ε' (epsilon) is the slant height of the cone.
The area of the base of the cone, which is a circle, is calculated using the formula B = πr². Therefore, the total surface area of the cone is the sum of these two areas.
In the given question, the radius 'r' is 3 ft, and the slant height 'ε' is 4 ft.
Substituting these values into the formulas gives us the lateral surface area as SA = π * 3 ft * 4 ft = 12π ft² and the area of the base as B = π * (3 ft)² = 9π ft².
Adding them together, the total surface area of the cone is SA + B = 12π ft² + 9π ft² = 21π ft².