posted by Barbara on .
An oil can is to have a volume 1000in^3 and is to be shaped like a sylinder with a flat bottom but capped by a hemisphere. Neglect the thickness of the material of the can and find the dimensions that will minimize the total amount of material needed to construct it.
Using Lagrange multipliers, you will need the following variables
Surface Area, A = πr^2+2πrh+2πr^2
You need to minimize A subject to V=1000 using the following objective function:
= πr^2+2πrh+2πr^2 + λ(V-1000)
find partial derivatives with respect to r, h, & λ to get 4 equations and solve for the system of equations in three unknowns r,h and λ.
Zh=2 %pi r + L %pi r^2 = 0
Eliminating L from first two equations gives r=h
Substitute h=r in third equation gives
Check my arithmetic