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Math Help - Optimization problem

  1. #1
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    Optimization problem

    The cost of manufacturing x number of pencils is given by: C = x^2 + 3x + 13
    If each pencil is sold for $5, maximize the profit.

    I thought profit was equal to the total amount made and subtract the cost of making them, so: profit = (5x) - (x^2 + 3x + 13) and then I differentiated and set it equal to 0 and found that x = 1 is the best to maximize the profit, but it's wrong. I think I set up the profit equation incorrectly. How do I do this?
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  2. #2
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    Re: Optimization problem

    Quote Originally Posted by PhizKid View Post
    The cost of manufacturing x number of pencils is given by: C = x^2 + 3x + 13
    If each pencil is sold for $5, maximize the profit.

    I thought profit was equal to the total amount made and subtract the cost of making them, so: profit = (5x) - (x^2 + 3x + 13) and then I differentiated and set it equal to 0 and found that x = 1 is the best to maximize the profit, but it's wrong. I think I set up the profit equation incorrectly. How do I do this?
    What, exactly, is the question? "Maximise the profit" is vague. You are correct that the maximum profit is when x= 1. Perhaps they want you state the actual profit in that case?
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  3. #3
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    Re: Optimization problem

    There is a possible twist to this problem. You have correctly calculated:

    P = (5x) - (x^2 + 3x + 13)=-x^2+2x-13

    \frac{dP}{dx}=-2x+2

    so x=1 is the maximum. But notice that P=-12 at this maximum. Surely it would be possible to close the pencil-making operation completely and thereby achieve a greater profit (or smaller loss, if you prefer) with P=0. Though, like HallsofIvy said, the question is vague. Perhaps they just want you to say that it's not possible to make a profit.

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