(Originais Teachy 2023) - Question Hard of Mathematics

An environmental engineer is designing a sprinkler irrigation system for a corn plantation in the shape of an inverted cone. The cone is 12m tall and the angle of the circular sector formed by the radius is 60º. To optimize the irrigation system, the engineer needs to know the total surface area of the cone, including the area of the base circle. The corn plant is circular, with a radius of 10m. Considering that the surface area of the cone is the sum of the areas of the base circle and the lateral surface, and that the formula for the lateral surface area of a cone is (pi * base radius * generatrix), where the generatrix is given by (square root of the square of the height plus the square of the base radius), calculate the total surface area of the cone so that the engineer can properly size the irrigation system. Consider pi as approximately 3.14 and use the formula for calculating the lateral surface area presented. Given the context presented, what would be the total surface area of the cone that the engineer should consider when designing the irrigation system for the inverted cone-shaped corn plantation?