calculus

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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
    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 -

    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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