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Math Help - 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,

    <br />
\begin{tabular}{ r l }<br />
\(2p + q\) &= \(60\) \\<br />
\(q\) &= \(60 - 2p\) \\ <br />
&= \(2(30 - p)\) \\<br />
\\<br />
\(Area\) &= \(p \times q\) \\<br />
\ &= \(2p(30 - p)\) \\<br />
\end{tabular}<br />

    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,

    <br />
\begin{tabular}{ r l }<br />
\(2p + q\) &= \(60\) \\<br />
\(q\) &= \(60 - 2p\) \\ <br />
&= \(2(30 - p)\) \\<br />
\\<br />
\(Area\) &= \(p \times q\) \\<br />
\ &= \(2p(30 - p)\) \\<br />
\end{tabular}<br />

    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 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 p=0,p=30 the midpoint is at p=15 this gives

    A(15)=2(15)^2=450

    or you can expand this out and use the vertex formula \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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