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Math Help - optimization problems

  1. #1
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    optimization problems

    The sum of the first number and twice the second number is 108 and the product is a maximum.
    So I did
    x + 2x = 108
    x+x=108/2 = 54
    2x=54
    x=54/2 = 27
    54,27 Im not sure if this is the right way to do it?
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  2. #2
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    Quote Originally Posted by Nightasylum View Post
    The sum of the first number and twice the second number is 108 and the product is a maximum.
    So I did
    x + 2x = 108
    x+x=108/2 = 54
    2x=54
    x=54/2 = 27
    54,27 Im not sure if this is the right way to do it?
    you called the second number x also ? did the problem tell you that the first and second numbers were equal?

    if not ...

    x + 2y = 108

    P = xy = (108 - 2y)y

    find the value of y that makes P a maximum.
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  3. #3
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    Quote Originally Posted by skeeter View Post
    you called the second number x also ? did the problem tell you that the first and second numbers were equal?

    if not ...

    x + 2y = 108

    P = xy = (108 - 2y)y

    find the value of y that makes P a maximum.
    Im a little lost on this can you show me step by step?
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  4. #4
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    Quote Originally Posted by Nightasylum View Post
    Im a little lost on this can you show me step by step?
    x+2y=108
    x=54-y
    54-y+2y=108
    y=108-54
    y=54
    2x+54=108
    2x=108-54
    x=27
    So x=27 y=54
    Like that?

    wait it wont work hmmm
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  5. #5
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    Quote Originally Posted by Nightasylum View Post
    x+2y=108
    x=54-y
    54-y+2y=108
    y=108-54
    y=54
    2x+54=108
    2x=108-54
    x=27
    So x=27 y=54
    Like that?

    wait it wont work hmmm
    the product is a maximum ...

    P = (108-2y)y

    you can solve this using calculus ... \frac{dP}{dy} = 0 ... and determine the value of y that maximizes P

    or, without calculus ... you can remember something about the graph of a quadratic function w/ a negative leading coefficient and where the maximum is located.
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