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Math Help - Maximising a quadratic function.

  1. #1
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    Maximising a quadratic function.

    Here is another one that I am having problems with.

    The cost (in dollars) of producing x items is

    C (x) = 4000 - 3x + (X^2 1000)

    If the items are sold for $4 each, find the value of x that maximises the Profit and find the maximum profit.


    I have figured out the equasion

    Profit = P = 4x - (4000- 3x + (X^2 1000)

    I tried to put it into my graphics calculator, and get the value for the max profit but its a linear equasion.

    Thanks in advance for any help

    P.S apologies in advance, I do not know how to put maths equasions into computers... maybe my next question should be about that!
    Last edited by mr fantastic; May 11th 2011 at 03:29 AM. Reason: Re-titled.
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  2. #2
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    You should know that since the x^2 term has a negative coefficient, that there will be a maximum turning point, which you find by completing the square to put the equation into turning point form...

    \displaystyle \begin{align*}P &=4x - \left(4000 - 3x + \frac{x^2}{1000}\right)\\ &= 4x - 4000 + 3x - \frac{x^2}{1000}\\ &= -\frac{x^2}{1000} + 7x - 4000\\ &= -\frac{1}{1000}\left(x^2 - 7000x + 4\,000\,000\right)\\ &= -\frac{1}{1000}\left[x^2 - 7000x + \left(-3500\right)^2 - \left(-3500\right)^2 + 4\,000\,000\right]\\ &= -\frac{1}{1000}\left[\left(x - 3500\right)^2 - 12\,250\,000 + 4\,000\,000\right]\\ &= -\frac{1}{1000}\left[\left(x - 3500\right)^2 - 8\,250\,000\right]\\ &= -\frac{1}{1000}\left(x - 3500\right)^2 + 8250\end{align*}

    So what is the maximum profit and how many units do they have to sell?
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  3. #3
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    Quote Originally Posted by jlee88 View Post
    Here is another one that I am having problems with.

    The cost (in dollars) of producing x items is

    C (x) = 4000 - 3x + (X^2 1000)

    If the items are sold for $4 each, find the value of x that maximises the Profit and find the maximum profit.


    I have figured out the equasion

    Profit = P = 4x - (4000- 3x + (X^2 1000)

    I tried to put it into my graphics calculator, and get the value for the max profit but its a linear equasion.
    Not with that [itex]x^2[/itex] in it!

    [quoteThanks in advance for any help

    P.S apologies in advance, I do not know how to put maths equasions into computers... maybe my next question should be about that![/QUOTE]
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