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Maximize the area

  1. Mar 26, 2009 #1
    1. The problem statement, all variables and given/known data
    A total of x feet of fencing is to form 3 sides of a level rectangular yard. What is the maximum possible area of the yard, in terms of x?

    3. The attempt at a solution
    So far I have 2 sides of the yard is denoted by y, and one side is denoted by z.
    Then, 2y+z=x
    We want to maximize the Area=yz

    Now, do I substitute one of the variables in the Area equation to get the answer? Then I know I need to differentiate, but with respect to what? I'm confused here.
     
  2. jcsd
  3. Mar 26, 2009 #2

    berkeman

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    Staff: Mentor

    You're on the right track. A=yz=y(substitute what for z?)

    Then you will have an equation for A in terms of y (the variable) and x (the constant). Since you can vary y to vary the area A, you will differentiate A with respect to y, and then do what to find the value of y that gives you maximum A?
     
    Last edited: Mar 26, 2009
  4. Mar 26, 2009 #3
    Ok, so I have
    A=yz=y(x-2y)=xy-2y^2
    A'=y-4y

    We want to find the critical points so set A'=0
    A'=y-4y=0
    y=4y

    ...that doesn't make sense...?
     
  5. Mar 26, 2009 #4

    berkeman

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    Staff: Mentor

    Your calculation of A' in the 2nd equation has one incorrect term in it. Remember what you are differentiating with respect to.

    [tex]\frac{d}{dy} (xy - 2y^2) = ?[/tex]
     
    Last edited: Mar 26, 2009
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