(Originais Teachy 2024) - Question Hard of Mathematics

An engineering company wants to build a conical silo for grain storage, aiming to optimize space and minimize the cost of the material used in the structure construction. The silo will have a height of 10 meters and the radius of the base will be variable, as the company intends to make silos of different sizes. Considering that the cost of the material used to build the lateral surface of a silo is proportional to the area of this surface and that the base area is not included in the cost, it is requested: 1) Develop an expression for the cost of the material as a function of the base radius 'r' and the height 'h' of the silo, knowing that the lateral area of a cone is given by 'A_cone = π*r*l', where 'l' is the generatrix of the cone. 2) Determine the base radius that minimizes the cost of the material needed to build the lateral surface of the silo.