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.
Lagrange Multipliers -
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