(Originais Teachy 2023) - Question Very Hard of Mathematics

An engineer is designing an irrigation system for a soccer field and decides that the shape of the system should be a cylinder, where the smaller base corresponds to the water inlet point and the larger base to the coverage area of the system. He needs to determine the amount of PVC pipes needed to connect all the water outlet points on the smaller base (one at each vertex of a regular polygon) to the inlet point on the larger base of the cylinder. The engineer knows that the smaller base has a regular polygon with 8 vertices and that the diagonal of the polygon measures 1 meter. In addition, the height of the cylinder is 5 meters. Using his knowledge of metric relationships in cylinders, calculate the distance that a PVC pipe must travel from the highest point of the smaller base to the inlet point on the larger base, justifying all steps of your reasoning.