A silo is to be constructed in a 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 as for the cylinder and the volume must be 730000 ft^3. if construction costs are to be minimized, whats should the radius be?
OK. Here is what I get. Not sure if it is correct but it looks like it might be. Say 'x' is cost per unit area for the cyclinder (Ac) then the cost per unit area for the half sphere (As) is 2.8x. You can then create an equation for cost (C) using this: C = 2.8*x*As + x*Ac Using equations for area and knowing that you only have half a sphere you expand this to. . . C = 2.8*x*(2*pi*r^2) + x*(pi*r^2 + 2*pi*r*h) You can find h with respect to r using the volume equations. 2/3*pi*r^3+pi*r^2*h = 730000 so h = (730000-2/3*pi*r^3)/(pi*r^2) The cost equation then becomes. . . C = 2.8*x*2*pi*r^2 + x*pi*r^2 + x*2*pi*r*(730000-2/3*pi*r^3)/(pi*r^2) Simplify and you should get. . . C = 16.547*x*r^2 + x*1460000/r Then find the derivative. . . dC/dr = 33.09*x*r - x*1460000/r^2 Find the min by setting the derivative to zero. . . dC/dr = 0 = 33.09*r - 1460000/r^2 1460000/r^2 = 33.09*r r^3 = 44122 r = 35.34 Hope you could follow that.
