Posted by **Barbara** on Saturday, November 5, 2011 at 6:24pm.

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 -
**MathMate**, Saturday, November 5, 2011 at 8:18pm
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.

- calculus -
**MathMate**, Saturday, November 5, 2011 at 8:44pm
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

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