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I'm haveing optimization problems

  1. Apr 6, 2004 #1
    This is homework (forgive me) but I don’t want an answer I would just like to know what I am doing wrong.

    Here is the problem:

    Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3cm and 4cm if tow dies of the rectangle lie along the legs.

    Here is what I did

    L = length of rectangle
    W = width
    Theta = angle 4 leg of the right triangle

    I’m trying to optimize W*L where :
    W < 3
    L < 4

    These are the equations I got:
    Tan(theta) = w/(4-L)
    Tan(theta) = (3-W)/L
    Tan(theta) = 3/4

    W = [itex] 3(4-L)/4 [/itex]


    {3 - 3(4-L)/4}/ L = 3/4

    and that is as far as this brain will take me...
    Last edited: Apr 7, 2004
  2. jcsd
  3. Apr 7, 2004 #2


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    Dearly Missed

    Basically, what you have done "wrong" is back-substituting, so that you end up with the trivially correct equation 3/4=3/4 .
    You already have a perfectly good expression for the width, w=3(4-L)/4.
    With this expression, what is the area of the rectangle?
    How can you find the maximum of this area?
  4. Apr 7, 2004 #3
    After you figure out that one, try this one.

    Basically the same problem, but one side of the rectangle is on the hypotenuse of the triangle.
  5. Apr 7, 2004 #4


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    Set up a coordinate system with the right angle at, (4,0) thus the line representing the hypotenuse is given by y= 3x/4.

    Let L and H be the sides of your rectangle. One corner must be on the line y(x) = 3x/4
    one side will be L=4-x the other H=y

    The area is A=L*H = (4-x)*y = (4-x)*3x/4

    Compute [tex] \frac {dA} {dx}= 0 [/tex]
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