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 2.8 times as much for the cylinder and the volume must be 710000 cubic feet. If construction costs are to be minimized, what should the radius be?
The volume of the silo is the volume of the cylinder + the volume of the hemisphere 71000 = pi x r^2 x h + (2/3) x pi x r^3 This can be rearranged to give h and a function of r h = (71000 - (2/3) x pi x r ^3)/(pi x r ^ 2) The cost of materials is the cost of the surface area of the cylinder (including a base) plus the surface area of the hemisphere (which coses 2.8 times as much per unit) C = (some constant) x (2.8 x 2 x pi x r^2 + pi x r ^2 + 2 x pi x r x h) you can replace h in this equation, which changes it to C = (some constant) x (6.6 x pi x r^2 + 142000/r - 4/3 x pi x r^2) The mininum cost happens when C' is 0. After taking the derivative you can divide by (some constant) to get rid of it and multiply both sides by r^2 to get a easy equation to solve. The answer will be the cube root of something...
