A silo (base not included) is to be constructed in the form of a cylinder surmounted by a hemisphere. The cost of construction per square unit of surface area is twice as great for the hemisphere as it is for the cylindrical sidewall. Determine the dimensions to be used if the volume is fixed and the cost of construction is to be kept to a minimum. Neglect the thickness of the silo and waste in construction.
surface area of top of silo is [4 pi r^2] / 2 surface area of a cylinder is 2 pi r h so we know that cost of top is 2 x cost of side [4 pi r^2] / 2 = 2 [2 pi r h] so 4 pi r^2 = 4 [2 pi r h] 4 pi r^2 = 8 pi r h pi r^2 = 2 pi r h r^2 = 2 r h r^2 / 2r = h r/2=h so I think the height is r / 2
