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Lagrange multiplier problem

  1. Nov 11, 2009 #1
    1. The problem statement, all variables and given/known data
    A window of fixed perimeter is in the shape of a rectangle surmounted by a semi-circle. Prove that its area is greatest when its breadth equals its greatest height.


    2. Relevant equations
    SA = lw + (pi*l^2)/4 <--- Thats what I got the surface area to be.
    Perimeter = 2w + L(1 + pi / 2)


    3. The attempt at a solution
    I can solve these problems with numbers, but when it comes to general problems I become unstuck. I tried using the same methord, but it didn't work. Some advice please?

    Thanks
     
  2. jcsd
  3. Nov 11, 2009 #2

    lanedance

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    Homework Helper

    Re: Optimisation

    you need to maximise the surface area with the perimeter constraint and the easiest way would be to use a lagrange multilpier..

    otherwise assume a constant value for the perimeter, say p, solve the perimeter eauqtion for l or w, then substitute back into the SA equation and minimise the function of (now) one variable
     
  4. Nov 12, 2009 #3
    Re: Optimisation

    Cheers, the second methord worked a treat! I'll look into the lagrange stuff, looks interesting :)
    Thanks once again.
     
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