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Math Help - need help

  1. #1
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    Exclamation Business Calculus - Need help! :(

    I've no idea is it the right section to post this,
    But I had some problems dealing with this problem.

    Total costs


    Market price


    Tax (per unit produced)


    Find:
    1. Quantity to produce to maximise profits
    2. What should t be, so as to maximise tax revenue?

    * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

    What I've done:

    I derived the profit function as:


    Then, I differentiate the profit function:



    Then I'm stuck cos' I can't find x.

    Last edited by geekie; December 15th 2008 at 05:30 PM.
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  2. #2
    Senior Member DeMath's Avatar
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    Quote Originally Posted by geekie View Post
    I've no idea is it the right section to post this,
    But I had some problems dealing with this problem.

    Total costs


    Market price


    Tax (per unit produced)


    Find:
    1. Quantity to produce to maximise profits
    2. What should t be, so as to maximise tax revenue?

    * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

    What I've done:

    I derived the profit function as:


    Then, I differentiate the profit function:



    Then I'm stuck cos' I can't find x.

    Conditional denotations:

    {\text{TC}}\left( x \right) is function of total costs
    {\text{P}}\left( x \right) is function of market price
    {\text{TR}}\left( x \right) is function of revenue
    {\text{T}}\left( x \right) is function of tax
    {\text{Pr}}\left( x \right) is function of profit


    \begin{gathered}{\text{TC}}\left( x \right) = \frac{7}{8}x^2  + 5x + 1 \hfill \\{\text{P}}\left( x \right) = 15 - \frac{3}{8}x{\text{ }} \Rightarrow {\text{ TR}}\left( x \right) = x \cdot {\text{P}}\left( x \right) = x\left( {15x - \frac{3}{8}x^2 } \right) = 15x - \frac{3}{8}x^2  \hfill \\{\text{T}}\left( x \right) = tx \hfill \\ \end{gathered}

    \begin{gathered}{\text{Pr}}\left( x \right) = {\text{TR}}\left( x \right) - {\text{TC}}\left( x \right) - {\text{T}}\left( x \right) =  \hfill \\= 15x - \frac{3}{8}x^2  - \left( {\frac{7}{8}x^2  + 5x + 1} \right) - tx = 10x - \frac{5}{4}x^2  - tx - 1 \hfill \\ \end{gathered}

    \begin{gathered}{\text{Pr}}\left( x \right) \to \max {\text{  if  }}\frac{d}{{dx}}{\text{Pr}}\left( x \right) \to 0 \hfill \\\frac{d}{{dx}}{\text{Pr}}\left( x \right) = \frac{d}{{dx}}\left( {10x - \frac{5}{4}x^2  - tx - 1} \right) = 10 - \frac{5}{2}x - t \hfill \\ \end{gathered}

    10 - \frac{5}{2}x - t = 0 \Leftrightarrow \frac{5}{2}x = 10 - t \Leftrightarrow 5x = 20 - 2t \Leftrightarrow \left[ \begin{gathered}x = 4 - \frac{2}{5}t \hfill \\t = 10 - \frac{5}{2}x \hfill \\ \end{gathered}  \right.
    Last edited by DeMath; January 1st 2009 at 06:04 PM.
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