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Maximizing the area

  1. Mar 14, 2009 #1
    1. The problem statement, all variables and given/known data
    A running track consists of a rectangle with a semicircle at each end. If the perimeter is to be exactly 440 yards, find the dimensions (x and r) that maximize the area of the rectangle. [Hint The perimeter is 2x + 2[tex]\pi[/tex]r



    2. Relevant equations



    3. The attempt at a solution
    Ok I attempted this twice and got the exact same answer, twice. Here is what I did.

    First I set up the equation: 440 = 2x + 2[tex]\pi[/tex]r

    I then set up the equation: Area (total) = [tex]\pi[/tex]r2 + 2rx where x is the length of the side of the field (not counting the semicircles) and r is the radius.

    I solved for x from the first equation and came up with x = 220 - [tex]\pi[/tex]r

    I then plugged the value of x into the second equation. Once I destrubuted it, I took the derivative and set it to zero and had 2r[tex]\pi[/tex] + 440 - 4[tex]\pi[/tex]r = 0

    Solving for r, I got 70.03. The answer in the back of the book is 110. What am I doing wrong?

    I appreciate the help!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 14, 2009 #2

    tiny-tim

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    Hi wcbryant87! :smile:
    erm :redface: … wrong area! :wink:
     
  4. Mar 14, 2009 #3
    haha. wow. the 'aha' moment has hit me. thanks!
     
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