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