What dimensions will minimize the cost of a tank with a volume of 5920 cubic feet, whose hemispherical ends cost twice as much as the sides per square foot of surface area?

Question

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. An industrial tank of this shape must have a volume of 5920 cubic feet. The hemispherical ends cost twice as much per square foot of surface area as the sides. Find the dimensions that will minimize the cost.

Answer 1

Here is the process. Volume is the volume of a cylinder plus the volume of a sphere.

V = πr²h + (4/3)πr³ = 5920 --> solve for h here

The cost of the material is the surface area of a cylinder not counting top and bottom plus twice the

surface area of a sphere

C = 2πrh + 2(4πr²) --> substitute what you found for h into this formula

Take the derivative of Cost with respect to r. Then set the derivative to zero and solve for r.

Once you have the value of the radius, r, use that to solve for the height, h.