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Thread: Optimization problem of a triangle inscribed in a circle

  1. #1
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    Exclamation Optimization problem of a triangle inscribed in a circle

    I need your help understanding how to go about the following problem,
    Triangle ABC has sides AB = BC. It is inscribed in a circle with center O, radius 10.0 cm. Find the value of the angle BAC that produces a maximum area for triangle ABC.
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  2. #2
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    Re: Optimization problem of a triangle inscribed in a circle

    a=AB=BC

    b=AC

    area = \frac{1}{2}a b \sin  A

    use the law of sines
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    Re: Optimization problem of a triangle inscribed in a circle

    I am supposed to find an equation, derivate it and then find theta. I am looking into that equation that you suggested, but i am stuck on how to draw it.
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    Re: Optimization problem of a triangle inscribed in a circle

    $Area = \dfrac{1}{2}(2y)(x+10) = y(x+10)$

    $\dfrac{dA}{dx} = y + (x+10) \cdot \dfrac{dy}{dx}$

    $x^2+y^2=10^2 \implies \dfrac{dy}{dx} = -\dfrac{x}{y}$ ...

    $\dfrac{dA}{dx} = y + (x+10) \cdot \left(-\dfrac{x}{y} \right) = y - \dfrac{x^2}{y} - \dfrac{10x}{y}$

    $y - \dfrac{x^2}{y} - \dfrac{10x}{y} = 0$

    $y^2 - x^2 - 10x = 0$

    $(10^2 - x^2) - x^2 - 10x = 0$

    $2x^2 + 10x - 100 = 0 \implies 2(x+10)(x-5) = 0 \implies x = 5$

    $m\angle{A} = \arctan\left(\dfrac{10+x}{y}\right)$
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    Re: Optimization problem of a triangle inscribed in a circle

    Thank you for your help. There is an easier way of solving it. If I put the sides in terms of sine and cosine and then differentiate. Your drawing is very helpful. ��
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    Re: Optimization problem of a triangle inscribed in a circle

    Quote Originally Posted by Dayli View Post
    Thank you for your help. There is an easier way of solving it. If I put the sides in terms of sine and cosine and then differentiate. Your drawing is very helpful. ��
    Using Skeeter's lovely diagram Area$(\Delta BCM)=A=0.5(x+10)(y)=0.5(x+10)(\sqrt{100-x^2})$
    You should be able to maximize $A$. Find the $m(\angle CBM)$. Double it to find $m(\angle CBA)$
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    Re: Optimization problem of a triangle inscribed in a circle

    Quote Originally Posted by Dayli View Post
    I am supposed to find an equation, derivate it and then find theta. I am looking into that equation that you suggested, but i am stuck on how to draw it.
    R=10

    a=2R \sin  A and b=2R \sin  B=2R \sin  (2A) since B=180-2A

    Therefore the area is given by

    \frac{1}{2}(2R \sin  A)* 2R \sin  (2A)*\sin  A

    =2R^2 \sin ^2A *\sin  (2A)

    differentiate and solve for A
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