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Maximizing area word problem

  1. Jan 24, 2012 #1
    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 of perimeter ≤ f+40, 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
    P=> 2x+y=60 => y=60-2x
    A=> xy=60x-2x^2

    P=> 2x+y=100 => y=100-2x
    A=> xy=100x-2x^2

    P=> 2x+y=160 => y=160-2x
    A=> xy=160x-2x^2

    3. The attempt at a solution
    I worked through the problem and found
    a. x=15, y=30 => A=450 sq ft
    b. x=25, y=50 => A=1250 sq ft
    c. x=40, y=80 => A=3200 sq ft

    I was just wondering if there was a way I could check these answers?
  2. jcsd
  3. Jan 24, 2012 #2


    Staff: Mentor

    There are at least a couple of ways, one of which doesn't use calculus. In each case your area function, A(x) has a graph that is a parabola that opens downward. The maximum area is attained at the vertex of the parabola. Complete the square to find the vertex.
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