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Math Help - Two shapes and one line.

  1. #1
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    Two shapes and one line.

    A piece of wire of length 60 inches is cut into two pieces. One piece is foldedinto a square and the other into an equilateral triangle. Find the maximum of the area of the square
    plus the area of the triangle. Choose the closest number from the list below.
    The furthest I've gotten is

    60 = 4x + 3y

    PS, this is just for a test I'm taking online, I'm not even in a calculus class. Don't think I'm trying to cheat or anything.
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  2. #2
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    Re: Two shapes and one line.

    Quote Originally Posted by Nervous View Post
    The furthest I've gotten is

    60 = 4x + 3y

    PS, this is just for a test I'm taking online, I'm not even in a calculus class. Don't think I'm trying to cheat or anything.
    Well if you cut the wire at length x, then the remaining length is 60 - x.

    If you make the square with the length 60 - x, then its side length is \displaystyle \begin{align*} \frac{60 - x}{4} \end{align*} and its area is \displaystyle \begin{align*} \left(\frac{60-x}{4}\right)^2 = \frac{3600 - 120x + x^2}{16} \end{align*}

    If the triangle is made with the length x, then its base is x and its height is \displaystyle \begin{align*} \frac{\sqrt{3}\,x}{2} \end{align*}, therefore its area is \displaystyle \begin{align*} \frac{\sqrt{3}\,x^2}{4} \end{align*}

    So the total area is \displaystyle \begin{align*} \frac{3600 - 120 + \left(4\sqrt{3} + 1\right)x^2}{16} \end{align*}

    Can you figure out how to find the maximum area now?
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    Re: Two shapes and one line.

    I derived the fraction and got:



    \frac{-120+2x+8\sqrt{3}x^2}{16}

    I set it equal to zero and got:

    x=~7.568

    And when I plugged that into the original equation, I got ~196.620

    But, the answer is supposed to be 225...
    Last edited by Nervous; October 2nd 2012 at 03:56 AM.
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    Lightbulb Re: Two shapes and one line.

    You equated area to zero! You should equate its derivative to zero in order to obtain x value that makes area maximum \frac{60}{1+4 \sqrt{3}} or something like that!
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    Re: Two shapes and one line.

    \frac{60}{1+4 \sqrt{3}} = ~7.568
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