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Math Help - Problem understanding question

  1. #1
    Senior Member Paze's Avatar
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    Problem understanding question

    I have a problem:

    1. Find two positive numbers whose sum is 300 and whose product is a maximum.

    What I don't understand is that last constraint: ''is a maximum''. What do they mean by a product being a maximum? Thanks.
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  2. #2
    Member agentmulder's Avatar
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    Re: Problem understanding question

    We have x + y = 300 and want to maximize x*y. From the first condition we get y = 300 - x. Now substitute.

    x(300 - x) = -x^2 + 300x

    This is a parabola that opens DOWNWARD therefore the maximum occurs at the vertex. The x co-ordinate of the vertex is given by

     x \ = \ -\frac{b}{2a} \ = \ -\frac{300}{-2}

    Can you continue?

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  3. #3
    Senior Member Paze's Avatar
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    Re: Problem understanding question

    Ah, I understand now what they mean.

    Can't I simply take the derivative of the substitution and set it to 0 to come to a conclusion? (That's what the class is all about at the moment).

    As so:

    \frac{d}{dx}-x^2+300x=-2x+300\\\\-2x+300=0\\\\x=150

    Which would imply that the largest number is given by

    150\cdot(300-150)=\\\\150\cdot150

    ? (btw can someone tell me why the first part of these spaces always has an indent?)

    Just one more question if this all checks out: Don't I have some specific range as well? I can't seem to put my finger on that range.
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  4. #4
    Member agentmulder's Avatar
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    Re: Problem understanding question

    Yes you can use derivative, wasn't sure if this was a calculus or pre-calculus question. The positive values for both x and y occur between the 2 roots of the parabola and nowhere else as a quick sketch will show so,

     0 \ < \ x \ < \ 300

    and

     0 \ < \ y \ < \ 300

    That range for y is a bit sneaky... if you pick x to range between 0 and 300 then y will range between 0 and 150 but i suppose the roles of x and y can be reversed without fear so technically you get both between 0 and 300.



    The indent may have something to do with your settings but i don't know what, i'm using the same code as you but i don't get that indent.

    P.S. If you're going to use calculus you will need to prove x = 150 gives the maximum by using the 1st derivative test.
    Last edited by agentmulder; May 13th 2013 at 12:26 AM.
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  5. #5
    Senior Member Paze's Avatar
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    Re: Problem understanding question

    Quote Originally Posted by agentmulder View Post
    Yes you can use derivative, wasn't sure if this was a calculus or pre-calculus question. The positive values for both x and y occur between the 2 roots of the parabola and nowhere else as a quick sketch will show so,

     0 \ < \ x \ < \ 300

    and

     0 \ < \ y \ < \ 300

    That range for y is a bit sneaky... if you pick x to range between 0 and 300 then y will range between 0 and 150 but i suppose the roles of x and y can be reversed without fear so technically you get both between 0 and 300.



    The indent may have something to do with your settings but i don't know what, i'm using the same code as you but i don't get that indent.

    P.S. If you're going to use calculus you will need to prove x = 150 gives the maximum by using the 1st derivative test.
    By 'first derivative test' do you mean this?

    f'(149)>0 , f'(151)<0 Therefore x=relative maxima.

    ?
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  6. #6
    Member agentmulder's Avatar
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    Re: Problem understanding question

    Yes, you are correct.

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  7. #7
    Senior Member Paze's Avatar
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    Re: Problem understanding question

    Quote Originally Posted by agentmulder View Post
    Yes, you are correct.

    Thank you very much!
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