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Math Help - Calculating Triangle Area Variable

  1. #1
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    Calculating Triangle Area Variable

    Bit of trouble differentiating this one. Can anyone help? Many thanks.

    Q.
    A curve has equation y = x2. Three points form the triangle apb. p(x,y) is a point on the curve, a, on the x-axis, is (6,0) & a final point b, also on the x-axis. bp \perp ab. (i) Express the coordinates of p, in terms of x only, (ii) Find the value of x if the area of the triangle abp is a maximum and hence find this maximum area.

    Attempt: (i) If y = x2, then p = (x, x2)
    (ii) If bp \perp ab, we can infer that b = (x,0).
    Area of a triangle apb on a plane is: \frac{1}{2}[(y2 - y1)(x1 - x3) - (x2 - x1)(y1 - y3)] => \frac{1}{2}[(x2 - 0)(6 - x) - (x - 6)(0 - 0)] => \frac{1}{2}[6x2 - x3]
    \frac{dA}{dx} = \frac{1}{2}(12x - 3x2) = 0 => 12x - 3x2 = 0 => 4x - x2 = 0 => x2 = 4x => x = 4

    Ans: (From text book): x = 2
    Last edited by GrigOrig99; April 10th 2012 at 02:32 PM.
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  2. #2
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    Re: Calculating Triangle Area Variable

    Your question says nothing about the triangle area...
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  3. #3
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    Re: Calculating Triangle Area Variable

    Apologies. I have re-edited post accordingly.
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  4. #4
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    Re: Calculating Triangle Area Variable

    If B is allowed to be located right of A, then there is no maximum triangle area. If B has to be left of A, then I agree that the maximum is reached when x = 4. When x = 2, the area is 8, but when x = 4, the area is 16.
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  5. #5
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    Re: Calculating Triangle Area Variable

    Thank you.
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