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 2.0 times as much as for the cylinder and the volume must be 840000 cubic feet. If construction costs are to be minimized, what should the radius be?
The surface area of the cylinder (including only one end) is AsubC = πr^2 + 2πrh (base plus side) The surface area of the hemisphere at the top is 1/2 the surface of a sphere with the same radius or: AsubHS = 2πr^2 Let c = cost per square foot for the cylinder, then Total cost = 2c2πr^2 + cπr^2 + c2πrh or TC = c(5πr^2 + 2πrh) Volume depends upon whether or not the hemisphere is included. The problem is considerably messier if it is included, so let's assume not. V = hπr^2 = 840000 ft^3 This allows us to solve for h in terms of r fairly simply giving h = 840000/πr^2 which may be substituted in the surface area cost formula to provide an expression for the surface area cost in terms of just the radius. or TC/c = 5πr^2 + 2πr(840000/πr^2) which simplifies to: TC/c = 5πr^2 + 1680000/r to find the minimum of this, differentiate wrt r giving (we will ignore the constant c, since we assume it is non-zero). TCprime(r) = 10πr - 1680000/r^2 solving for 0 = 10πr - 1680000/r^2 gives 1680000/r^2 = 10πr or 53503.18 = r^3 cubert(53503.18) = 37.68ft. = r Provide h by substituting back into the equation for h in terms of r. This yields h = 188.42 ft. I have tried to be careful with this, but cannot be absolutely certain that I have not goofed someplace on the algebra or calculations. I strongly suggest that you review this carefully yourself. If you decide that the volume of the hemisphere on top is included, the volume becomes: V = 2πr^3/3 + hπr^2 So the expression for h is not quite so simple.
I was given this problem on a final exam as a freshman. You need to minimize the surface ares. You know that the area of the top is (Pi)*r^2 and the area of the cylinder portion is 2*(Pi)*r*h where r is the radius and h is the height. At this point, it looks like there are two unknowns. This is where you bring in the volume. You vet: V = (Pi)*r^2 * h So: 840,000 = Pi)*r^2 * h You can use algebra to get 'h' in terms of r and go back to the first equation. This puts surface area as a function of r. Take the first derivative and solve for r
