A silo is to be constructed in the form of a cylinder(only 1 of 2 bases included) topped by a hemisphere. The construction cost per square unit of surface area for the hemisphere is 1.5 times as much for the cylinder and the volume must be 900,000 cubic feet. If construction costs are to be minimized, what should the radius be?I keep getting 43.56 feet, but that's wrong.
I find r = 15 * cubic-root of (100/pi) = 47.538... feet The volume is V = pi r? h + 2/3 pi r? = 900000 The area of the cylinder is A1 = 2pi r h + 2 pi r? The area of the hemisphere is A2 = 2pi r? If c is the cost per square unit, then the total cost is : C(r) = c*A1 + 1.5c*A2 = 2pi c (r h + 2r?) h = 900000/ (pi r?) - 2/3 r thus C(r) = 2pi c (900000/(pi r) + 4/3 r?) Derivating : C'(r) = 2pi c (- 900000/(pi r?) + 8/3 r) The minimum is when C'(r) = 0 8/3 r = 900000 / (pi r?) r? = 2700000 / (8 pi) = 15? (100/pi) r = 15 * (100/pi)^(1/3) = 47.538... feet
