Question

Optimization: An industrial tank formed by adjoining a hemisphere to each end of a right circular cylinder has a volume of 3000 cubic feet. If the construction cost of the hemispherical ends is twice as much per square foot of surface area as the sides, find the dimensions that will minimize cost.

Answer 1

Area of cylinder=2 pi r h
Area of sphere (both ends) = 4 pi r^2

If cost per sq ft is k then total cost C = 2k pi r h + 8k pi r^2

To find the minimum - differentiate (dC/dr) and set to 0
This gives you the ratio of r/h for minimum cost

Total vol - 3000 - so using r/h you can calculate the cylinder dimensions for min cost

Answer 2

CORRECTION

Area of cylinder=2 pi r h
Area of sphere (both ends) = 4 pi r^2

If cost per sq ft for cylinder is k, then cost per sq ft of the sphere is 2k
Therefore total cost C = 2k pi r h + 8k pi r^2

To find the minimum - differentiate (dC/dr) and set to 0
This gives you the ratio of r/h for minimum cost

Total vol - 3000 - so using r/h you can calculate the cylinder dimensions for min cost