(Originais Teachy 2023) - Question Very Hard of Mathematics

A cosmetics industry wants to manufacture a cylindrical perfume bottle for its new launch. The bottle should have a height of 10 cm and a volume of 500 cm³. Additionally, the factory has decided that the total surface area of the bottle, including the base and the top, should be minimized to reduce the amount of material used. Considering that the base area of the cylinder is directly proportional to the cylinder's volume and that the total surface area (A) of a cylinder with radius (r) and height (h) is given by A = 2 * pi * r * (r + h), calculate the radius (in cm) that minimizes the total surface area of the bottle. After finding the value of the radius that meets this requirement, check if the dimensions (height and radius) of the proposed bottle simultaneously meet the volume and total surface area conditions established by the industry. If the dimensions do not meet both requirements, discuss why this occurs and suggest adjustments that could be made to optimize the bottle shape, while keeping the height constant at 10 cm.