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Homework Help: PLEASE HELP maximizing area problem

  1. Jan 25, 2012 #1
    PLEASE HELP!! maximizing area problem

    1. The problem statement, all variables and given/known data
    1. The problem statement, all variables and given/known data
    A farmer wants to build a rectangular pen. He has a barn wall 40 feet long, some or all of which must be used for all or part of one side of the pen. In other words, with f feet of of fencing material, he can build a pen with a perimeter of up to f+40 feet, and remember he isn't required to use all 40 feet.
    What is the maximum possible area for the pen if:
    a. 60 feet of fencing material is available
    b.100 feet of fencing material is available
    c. 160 feet of fencing material is available


    2. Relevant equations
    a.
    P=> 2x+y=60 => y=60-2x
    A=> xy=60x-2x^2


    3. The attempt at a solution
    I worked through the problem and found
    a. For 60 ft of fencing material
    x=15, y=30 => A=450 sq ft

    I am not sure how to do b and c.
    This is my attempt, but I know it is wrong (slightly)
    b. For 100 ft of fencing material,
    P=> 2x+y=100 => y=100-2x
    A=> xy=100x-2x^2
    working through this, I end up with
    x=25, y=50 => A=1250 sq ft

    I know this is wrong because the barn wall is only 40 feet long, and y=50, but all of the 100 feet of fencing has been used up by 2x+y, so there will be 10 feet of fencing missing. How do I rework the problem? Please help i've been working on this for hours!! :(
     
  2. jcsd
  3. Jan 25, 2012 #2

    vela

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    Re: PLEASE HELP!! maximizing area problem

    The variable y is the one with the constraint on it. It has to lie somewhere in the interval [0, 40]. Write the area as a function of y and find where it attains its maximum on that interval.
     
  4. Jan 25, 2012 #3
    Re: PLEASE HELP!! maximizing area problem

    This is linear programming problem. With following equations for b

    [tex]x\geqslant 0\quad ; 0\leqslant y\leqslant 40[/tex]

    [tex]2x+2y \leqslant 140[/tex]

    Maximize the function [itex]A=xy[/itex]. Use LP theorem
     
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