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Explain why this is correct (Optimization Problem)

  1. Mar 29, 2016 #1
    1. The problem statement, all variables and given/known data
    A piece of wire, 100 cm long, needs to be bent to form a rectangle. Determine the dimensions of a rectangle with the maximum area.

    2. Relevant equations
    P = 2(l+w)
    A = lw

    3. The attempt at a solution
    This is what I don't understand, the solutions that I saw from looking around is:

    l = x
    w = (100 -2x) / 2

    I don't understand why is width portrayed as shown above, and why the length is also potrayed as above, the solution goes onto:

    A = (x)(100 - 2x / 2)
    A = (x)(50 = x)
    A = 50x - x^2
    A prime = 50 - x^2
    Insert 0 for A prime
    0 = 50 - x^2
    x=25
    With therefore means the length and width are 25cm.

    I understand the algebra, I do not understand how to get the length and width equation, wondering if someone could explain it to me.
     
  2. jcsd
  3. Mar 29, 2016 #2

    Mark44

    Staff: Mentor

    The 100 cm of wire will be the perimeter of the rectangle. From your formula for P, solve for the width w in terms of l. Also, introducing x as a variable is more complicating than just using w and l.
    The first equation above is not written correctly. The second group in parentheses is 100 - 2x/2, which is properly interpreted as ##100 - \frac {2x} 2= 100 - x##. Obviously that's not what you meant, so the equation should have been written as A = x(100 - 2x)/2.

    The second equation has a typo -- you wrote = instead of -.
     
  4. Mar 29, 2016 #3

    Simon Bridge

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    Homework Helper
    Gold Member
    2016 Award

    Forget about the characteristic of length and width... a rectangle has two pairs of parallel sides... one pair has length x and the other has length y. Note the special case that x=y is allowed even though we said "rectangle".
    If y>x then y is the length. If x>y then x is the length... but it does not matter to the maths, we can focus on either.

    At this stage the x and y aee just labels... the next step uses maths to describe how these are related to the area and the perimeter. If the area is A and the perimeter is p, write down the equations for these in terms of x and y.

    You need an ewuation for area in terms of only one other variable... so pick x or y, doesn't matter which, and make A depend only on that.

    Note. Your final working contains two errors which cancel each other out... check the derivative.
     
    Last edited: Mar 29, 2016
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