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Math Help - differentiation: maxima and minima

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    differentiation: maxima and minima

    p80 q11 ex18c
    question:
    A and D are two towns on the same side of the main road BC. A road APD is to be made to join A and D to the main road. where should the location of P be in order that the length of the road is its smallest? you can only use differentiation method.

    my working:
    let l be the length of the road, x be the distance of BP
    l = \sqrt {x^2+4} +\sqrt {(10-x)^2+9}
    dl/dx = \frac 1 2 (x^2+4)^{-\frac 1 2 } (2x) + \frac 1 2 (x^2 -20 x +109)^{-\frac 1 2 } (2x-20)
    = x(x^2+4)^{-\frac 1 2 }+ (x-10)(x^2 - 20 x +109)^{-\frac 1 2 }
    i cannot find the value of x when dl/dx = 0 . thanks!
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    Hello,

    Quote Originally Posted by afeasfaerw23231233 View Post
    p80 q11 ex18c
    question:
    A and D are two towns on the same side of the main road BC. A road APD is to be made to join A and D to the main road. where should the location of P be in order that the length of the road is its smallest? you can only use differentiation method.

    my working:
    let l be the length of the road, x be the distance of BP
    l = \sqrt {x^2+4} +\sqrt {(10-x)^2+9}
    dl/dx = \frac 1 2 (x^2+4)^{-\frac 1 2 } (2x) + \frac 1 2 (x^2 -20 x +109)^{-\frac 1 2 } (2x-20)
    = x(x^2+4)^{-\frac 1 2 }+ (x-10)(x^2 - 20 x +109)^{-\frac 1 2 }
    i cannot find the value of x when dl/dx = 0 . thanks!
    \frac{dl}{dx}=\frac{x}{\sqrt{x^2+4}}+\frac{x-10}{\sqrt{x^2-20x+109}}

    with x between 0 and 10.

    Put on the same denominator :

    \frac{dl}{dx}=\frac{x \sqrt{x^2-20x+109}+(x-10) \sqrt{x^2+4}}{\sqrt{x^2-20x+109} \sqrt{x^2+4}}

    The denominator never annulates, so it's ok;

    x \sqrt{x^2-20x+109}+(x-10) \sqrt{x^2+4}=0

    x \sqrt{x^2-20x+109}=(10-x) \sqrt{x^2+4}

    Squaring the equation :

    x^2 (x^2-20x+109)=(100-20x+x^2)(x^2+4)

    {\color{blue}x^4}{\color{red}-20x^3}+109x^2=100x^2+400{\color{red}-20x^3}-80x+{\color{blue}x^4}+4x^2

    109x^2=100x^2+400-80x+4x^2

    x^2+16x-80=0

    Can you solve it from here ? (don't forget that x is between 0 and 10)
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