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 1.6 times as much for the cylinder and the volume must be 850000 cubic feet. If construction costs are to be minimized, what should the radius be?I don't even know how to make an equation describing it, how do I solve this?
Cost = 1* [ 2 π r h] + 1.6*[2π r?] + [ ?] 850000 = π r? h + [2/3]π r^3 { solve this for h and put into the cost function}....if the base is to be included then ? = 1* π r ?, otherwise ? = 0...now YOU should be able to minimize the cost function
A_cyl = pi r^2 + 2 pi r H A_cap = 2 pi r^2 Cost = Cost_cyl * (1.6 * A_cap + A_cyl) V_cyl = pi r^H V_cap = 2/3 pi r^3 V = V_cap + V_cyl = 850000 ft^3 ===== Minimize cost: 1) Write Cost(r) by substituting Cost and area eqns above. 2) Use the volume relationships to eliminate H from the expression. 3) Compute d(Cost)/dr 4) Find r when d(Cost)/dr = 0 5) Reject the solution r=0.
