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Applied Optimization problem

  1. Nov 20, 2007 #1
    1. The problem statement, all variables and given/known data

    Find the area of the largest rectangle that can be inscribed in the region bounded by the graph of y = (4-x)/(2+x) and the coordinate aces in the first quadrant.

    I think my only problem with this one is taking the derivative,

    this is what i get y' = (-x^2 - 4x + 8)/(2+x)^2

    Critical numbers: [1-root(48)]/-2, but that doesn't seem to be giving me a maximum value, can someone take a second look this.
  2. jcsd
  3. Nov 20, 2007 #2
    If I'm not mistaken:

    A = xy, x = x, y = (4-x)/(2+x)

    A = x(4-x/2+x)
    = (4x - x^2 / 2 + x)

    Using Quotient Rule (feel free to use the product rule if you want to):
    dy/dx = ((2 + x)(4 -2x) - (4x - x^2)(1)) / (2 + x)^2

    =8 -4x +4x -2x^2 - 4x + x^2 / g^2
    dy/dx = 8 -x^2 -4x / 4 + 4x + x^2

    Graphing that, I can see 2 roots.
    (can't be bothered actually solving properly for them now though, :) )
    They are:
    -5.46410 & 1.464101

    Using 1.464101 leads to an area of about 1.0717
    Graphing the original function to be optimized, I see that this is correct.
  4. Nov 20, 2007 #3
    Okay, doing a poly long divide, I get the dy/dx as:
    -1 + 12/(x+2)(x+2) = 0
    x^2 + 4x - 8 = 0

    This is interestingly the numerator of the original function.

    Solving this with quadratic formula yields the same results.
  5. Nov 21, 2007 #4
    my error was in using the quadratic formula, i took -a instead of -b which is why i got a strange value.
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