A silo (base not included) is to be constructed in the form of a cylinder surmounted by a hemisphere. The cost of construction per square unit of surface area is twice as great for the hemisphere as it is for the cylindrical sidewall. Determine the dimensions to be used if the volume is fixed and the cost of construction is to be kept to a minimum. Neglect the thickness of the silo.I am having trouble getting a cost function in terms of a single variable.
let r be the radius of the hemisphere and the cylinder, h the height of the cylinder. the volume of silo V = pi*r^2*h + 2/3 * pi*r^3 so h = (V - 2/3 * pi*r^3)/(pi*r^2) (1) the surface area of the hemisphere A1 = 2*pi*r^2 the surface area of the cylindrical sidewall. A2 = 2*pi*r*h The cost C = k*(2*A1 + A2) where k is a constant. C = k*(2*2*pi*r^2 + 2*pi*r*h) (2) use eq.(1) to replace h in eq.(2) we get C is a function of a single variable r.
