Question

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.

Answer 1

Using Lagrange multipliers, you will need the following variables
radius, r
height, h
Volume, V=πr^2h+(2/3)πr^3=1000
Surface Area, A = πr^2+2πrh+2πr^2
λ=Lagrange multiplier
You need to minimize A subject to V=1000 using the following objective function:

Z(r,h,V,λ)
= 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.

Answer 2

Partial derivatives:
Zr=3r+h+L(r^2+rh)=0
Zh=2 %pi r + L %pi r^2 = 0
ZL=πr^2h+(2/3)πr^3-1000 =0

Eliminating L from first two equations gives r=h
Substitute h=r in third equation gives
r=(600/%pi)^(1/3)
=5.7588 approx.

Check my arithmetic