A silo(base not included) is to be constructed in the form of a cylinder surrounded by a hemisphere . The cost of construction per square unit of surface area is three times as great for the hemisphere as it is for the cylindrical sidewall.Determine the dimensions to be used if the volume is fixed at 4000 cubic units and the cost of construction is to be kept to a minimum . neglect the thickness of the silo and waste in constructionround to the nearest tenth as needed
I assume that by surrounded you mean capped. In that case, let r be both the radius of the cylinder and of the sphere and h be the height of the cylinder. The surface area of the cylinder is 2pirh and of the sphere is 1/2(4pir^2) = 2pir^2. V = pi(r^2)h + 1/2(4/3pir^3). 4000 = pi(r^2)h + 1/3pir^3. h = [4000 - 1/3pir^3]/pir^2. Let C = total cost and c = square foot cost of the cylinder. C = c(2pirh) + 3c(2pir^2). Now replace h with its r expression and minimize dS/dr
