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Math Help - Inequalities

  1. #1
    ADY
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    Inequalities

    Hi guys,

    ok, lets hope you can help me out!


    x and y must lie within or on the boundary of shaded region of my drawing

    * What are the two constraints on the problem?

    * Whats the max. value of the f 3x + 6y within the shaded region?

    Thankyou!
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  2. #2
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    Quote Originally Posted by ADY View Post
    Hi guys,

    ok, lets hope you can help me out!


    x and y must lie within or on the boundary of shaded region of my drawing

    * What are the two constraints on the problem?

    * Whats the max. value of the f 3x + 6y within the shaded region?

    Thankyou!
    If the points of the borderlines belong to the shaded region too then the shaded region is described by:

    y\leq-\frac52 x + 10~\wedge~y\leq -x+6~\wedge~x\geq 0~\wedge~y\geq 0

    If f = 3x+6y~\implies~y=-\frac12 x+\frac16 f this line has to pass at least through one point of the shaded region. f gets a maximum if the line is translated as much up as possible.
    Thus f is maximal if x = 0 and y = 6. In this case f = 36.
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  3. #3
    ADY
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    Quote Originally Posted by earboth View Post
    If the points of the borderlines belong to the shaded region too then the shaded region is described by:

    y\leq-\frac52 x + 10~\wedge~y\leq -x+6~\wedge~x\geq 0~\wedge~y\geq 0
    So two options that represent the problem could be shown with what 2 equations?


    Would these be two options that represent a constraint?

    y+\frac52 x \geq 10 ?

    y\leq\frac52 x - 10 ?


    or

    y+\frac52 x \leq 10 ?

    y\geq 10 + \frac25x ?
    Last edited by mr fantastic; February 15th 2009 at 07:43 PM. Reason: Merged posts
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  4. #4
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    Quote Originally Posted by ADY View Post
    So two options that represent the problem could be shown with what 2 equations?


    Would these be two options that represent a constraint?

    y+\frac52 x \geq 10 ?

    y\leq\frac52 x - 10 ?


    or

    y+\frac52 x \leq 10 ?

    y\geq 10 + \frac25x ?
    I'm not sure if I understand you correctly...

    Have a look at the graph which I posted with my previous reply:

    y\leq -x+6,\ if\ 0\leq x < \frac83~\wedge~\frac{10}3\leq y \leq 6 .... and ..... y\leq \frac52 x + 10 ,\ if\ \frac83 \leq x \leq 4 ~\wedge~0 \leq y \leq \frac83

    Since the line y = -\frac12 x+\frac16 f has to pass through the highest point of the region build by the constraints, so that the value of f becomes a maximum, you have to look for this "highest" point. You find it in the first part of the constraints.
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