(Originais Teachy 2024) - Question Medium of Mathematics

Consider a right circular cone with height 'h' and base radius 'r'. When we cut the cone with a plane parallel to the base, we obtain a figure known as a 'frustum' or 'cone trunk'. Suppose we want to build a water reservoir in the shape of a cone trunk, where the area of the smaller base section is 'A' and the area of the larger base section is 'B', with 'A' < 'B'. The reservoir must have a capacity of 'C' cubic meters of water. (1) Prove that the height 'H' of the cone trunk can be expressed as a function of 'A', 'B', and 'h'. (2) Calculate the height 'H' in terms of 'A', 'B', 'h', and 'C' and determine the shapes that the cross-sections must have to ensure the desired amount of water, considering 'h', 'r', 'A', and 'B' as independent variables.