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Math Help - What is the largest area of an isoceles triangle with

  1. #1
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    What is the largest area of an isoceles triangle with

    What is the largest area of an isosceles triangle with the perimeter 1 cm?
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  2. #2
    Junior Member Infophile's Avatar
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    Hello,

    I find \color{red}\frac{\sqrt{3}}{36} cm.

    I give my solution after diner.

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  3. #3
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    Quote Originally Posted by gtgamer140@yahoo.com View Post
    What is the largest area of an isosceles triangle with the perimeter 1 cm?
    Hi

    Let a be the equal length of 2 sides and b the length of the 3rd side.

    The perimeter is p = 1 = 2a+b

    The are is given by the formula S = \sqrt{\frac{p}{2}\left(\frac{p}{2}-a\right)\left(\frac{p}{2}-a\right)\left(\frac{p}{2}-b\right)}

    S = \sqrt{\frac{1}{2}\left(\frac{1}{2}-\frac{1-b}{2}\right)\left(\frac{1}{2}-\frac{1-b}{2}\right)\left(\frac{1}{2}-b\right)}

    S = \sqrt{\frac{1}{2}\:\frac{b}{2}\:\frac{b}{2}\:\frac  {1-2b}{2}}

    S = \frac{b}{4}\:\sqrt{1-2b}

    Set the derivative of S to 0 to find b_max then S_max
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  4. #4
    Junior Member Infophile's Avatar
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    Exactly,

    But it's not necessary to know this formula.

    You have h^2=a^2-\left(\frac{b}{2}\right)^2 with b=1-2a and S=\frac{bh}{2}.

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