Math

The equation P=240I-8I^2 represents the power, P, (in watts) of a 240 volt circuit with a resistance of 8 ohms when a current of I amperes is passing through the circuit.

Find the maximum power (in watts)that can be delivered in this circuit.

I have trouble to find the maximum power,please help! THANKS A LOT!

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  1. That is a parabola.
    If you know calculus, set dP/di = 0

    I will continue assuming you do not know calculus and finding the vertex of the parabola by completing the square

    - 8 i^2 + 240 i = P

    i^2 - 30 i = - P/8

    i^2 - 30 i +(30/2)^2 = -P/8 + 225

    (i-15)^2 = -P/8 + 225

    vertex at i = 15 and P at 225*8 = 1800

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  2. From this part on, I don't really get it:
    i^2 - 30 i +(30/2)^2 = -P/8 + 225

    (i-15)^2 = -P/8 + 225

    vertex at i = 15 and P at 225*8 = 1800

    Can you please explain???

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  3. I solved that quadratic equation by "completing the square".
    when you have the equation in the form:

    1 x^2 + b x = -c

    take half of b and square it
    (b/2)^2 in this case (30/2)^2 = 15^2 = 225

    add that to both sides
    1 x^2 + b x + (b/2)^2 = - c + (b/2)^2

    the left side factors
    (x + b/2)(x + b/2) = -c + (b/2)^2
    in our case b/2 = 15 so (b/2)^2 = 225
    or
    (x+b/2)^2 = -c + (b/2)^2
    in our case
    (x-15)^2 = -P/8 + 15^2

    take sqrt of both sides remembering that - the square root is also a solution.
    x+b/2 = +/- sqrt( b^2/4 - 4c/4)

    x = (- b/2 +/- (1/2)sqrt ( b^2 - 4 c)

    which you will recognize as the quadratic equation with a = 1
    We fixed a as one by dividing by a right at the start.

    The main thing is that -b/2a or in our case -b/2 is the axis of symmetry of the parabola. If the parabola faces up (holds water), the minimum is there. If the parabola faces down ( sheds water - our case), the maximum is there.

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