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Thread: Length and breadth of rectangle from perimeter

  1. #1
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    Length and breadth of rectangle from perimeter

    Hi,

    Looking for a hint on this one. Right now a bit clueless on how to proceed.



    ABCD is a rectangular piece of land and XY is a wall. A fence 60 meters long is used to fence up the land. Find the length and the breadth of the rectangular piece of land when the area is maximum.

    I denoted AB = p and BC = q.

    Given,

    $\displaystyle
    \begin{tabular}{ r l }
    \(2p + q\) &= \(60\) \\
    \(q\) &= \(60 - 2p\) \\
    &= \(2(30 - p)\) \\
    \\
    \(Area\) &= \(p \times q\) \\
    \ &= \(2p(30 - p)\) \\
    \end{tabular}
    $

    Am I on the right track. I don't see any other data to use here. Can you please provide a hint on how to proceed.

    Thanks!
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by mathguy80 View Post
    Hi,

    Looking for a hint on this one. Right now a bit clueless on how to proceed.



    ABCD is a rectangular piece of land and XY is a wall. A fence 60 meters long is used to fence up the land. Find the length and the breadth of the rectangular piece of land when the area is maximum.

    I denoted AB = p and BC = q.

    Given,

    $\displaystyle
    \begin{tabular}{ r l }
    \(2p + q\) &= \(60\) \\
    \(q\) &= \(60 - 2p\) \\
    &= \(2(30 - p)\) \\
    \\
    \(Area\) &= \(p \times q\) \\
    \ &= \(2p(30 - p)\) \\
    \end{tabular}
    $

    Am I on the right track. I don't see any other data to use here. Can you please provide a hint on how to proceed.

    Thanks!
    You are right where you need to be!

    The quadratic $\displaystyle A(p)=2p(30-p)=-2p^2+60p$ maximum can be found in a few different ways.

    1st symmetry the vertex must be at the midpoint of the p-intercepts. Since $\displaystyle p=0,p=30$ the midpoint is at $\displaystyle p=15$ this gives

    $\displaystyle A(15)=2(15)^2=450$

    or you can expand this out and use the vertex formula$\displaystyle \displaystyle p=\frac{-b}{2a}=\frac{-60}{-4}=15$
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  3. #3
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    Thanks @TheEmptySet. Guess I blanked out at the final dash. I didn't simplify the expression and just didn't think the problem through. And didn't know about the vertex formula. You explained it really well. Thanks a lot. Learned some great new stuff today!
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