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Thread: Help with this problem please

  1. #1
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    Help with this problem please

    $\displaystyle \\ \mbox{Give that,} \\ x_A , x_B >1 \\ \mbox{ and } \\ Y_A,Y_B\geqslant0 \mbox{ (but strictly positive for at least one)}$
    $\displaystyle Y_A + Y_B = 100$

    $\displaystyle P_A=Y_A(x_A -1) - Y_B$
    $\displaystyle P_B=Y_B(x_B -1) - Y_A$
    Problem:
    $\displaystyle x_A \mbox{ and } x_B \mbox{ is given } $
    $\displaystyle \mbox{Find values of }Y_A, Y_B \mbox{ such that:}$
    1) Both $\displaystyle P_A$ and $\displaystyle P_B$ is always greater than zero. Are there any restictions needed on $\displaystyle x$ for this to be true?

    ================================================== ========
    EDIT (21 OCT 07):

    Let me clarify what my actual question is. Sorry for not making this clear enough early on.

    The question I am asking is this:
    1) What restrictions (i.e. the general rule) are needed on $\displaystyle x_A $ and $\displaystyle x_B$ so that there are some range of values of $\displaystyle Y_A$ and $\displaystyle Y_B$ to make both $\displaystyle P_A$ and $\displaystyle P_B$ positive?
    2) What is the restriction (i.e. the general rule) for $\displaystyle x_A$ and $\displaystyle x_B$ such that no matter what combination of $\displaystyle Y_A$ and $\displaystyle Y_B$ is chosen (but still sums to 100), $\displaystyle P_A$ and $\displaystyle P_B$ is always negative?
    3) How can you prove your answers to questions 1) and 2) ?
    Last edited by tongzilla; Oct 21st 2007 at 01:03 AM.
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  2. #2
    Super Member angel.white's Avatar
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    Quote Originally Posted by tongzilla View Post
    $\displaystyle \\ \mbox{Give that,} \\ x_A , x_B >1 \\ \mbox{ and } \\ Y_A,Y_B\geqslant0 \mbox{ (but strictly positive for at least one)}$
    $\displaystyle Y_A + Y_B = 100$

    $\displaystyle P_A=Y_A(x_A -1) - Y_B$
    $\displaystyle P_B=Y_B(x_B -1) - Y_A$
    Problem:
    $\displaystyle x_A \mbox{ and } x_B \mbox{ is given } $
    $\displaystyle \mbox{Find values of }Y_A, Y_B \mbox{ such that:}$
    1) $\displaystyle P_A$ and $\displaystyle P_B$ is always greater than zero. Are there any restictions needed on $\displaystyle x$ for this to be true?
    I think you could do it like this:

    $\displaystyle Y_A=Y_B=50$
    Which satisfies $\displaystyle Y_A + Y_B = 100$

    Then $\displaystyle x_A, x_B > 2$

    Which satisfies:
    $\displaystyle P_A=Y_A(x_A -1) - Y_B$
    because Y_A = Y_B
    $\displaystyle P_A=Y_A(x_A -1) - Y_A$
    $\displaystyle P_A=Y_A[(x_A -1) - 1]$
    $\displaystyle P_A=Y_A(x_A -2)$
    And because we define $\displaystyle x_A>2$ we know that $\displaystyle x_A-2>0$ which means that $\displaystyle Y_A(x_A-2)>0$, which means that $\displaystyle P_A > 0$

    And the math will be the same for the other equation as well.
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  3. #3
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    EDIT (21 OCT 07):

    Let me clarify what my actual question is. Sorry for not making this clear enough early on.

    The question I am asking is this:
    1) What restrictions (i.e. the general rule) are needed on $\displaystyle x_A $ and $\displaystyle x_B$ so that there are some range of values of $\displaystyle Y_A$ and $\displaystyle Y_B$ to make both $\displaystyle P_A$ and $\displaystyle P_B$ positive?
    2) What is the restriction (i.e. the general rule) for $\displaystyle x_A$ and $\displaystyle x_B$ such that no matter what combination of $\displaystyle Y_A$ and $\displaystyle Y_B$ is chosen (but still sums to 100), $\displaystyle P_A$ and $\displaystyle P_B$ is always negative?
    3) How can you prove your answers to questions 1) and 2) ?
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  4. #4
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    Quote Originally Posted by angel.white View Post
    I think you could do it like this:

    $\displaystyle Y_A=Y_B=50$
    Which satisfies $\displaystyle Y_A + Y_B = 100$

    Then $\displaystyle x_A, x_B > 2$

    Which satisfies:
    $\displaystyle P_A=Y_A(x_A -1) - Y_B$
    because Y_A = Y_B
    $\displaystyle P_A=Y_A(x_A -1) - Y_A$
    $\displaystyle P_A=Y_A[(x_A -1) - 1]$
    $\displaystyle P_A=Y_A(x_A -2)$
    And because we define $\displaystyle x_A>2$ we know that $\displaystyle x_A-2>0$ which means that $\displaystyle Y_A(x_A-2)>0$, which means that $\displaystyle P_A > 0$

    And the math will be the same for the other equation as well.
    So is x_A and x_B > 2 the one and only restriction needed? Does it only work for Y_A and Y_B =50 or it also works for other values?
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