Results 1 to 2 of 2

Math Help - Maximum profit

  1. #1
    Newbie
    Joined
    Apr 2008
    Posts
    16

    Maximum profit

    find the price per unit p that produces the max profit P.

    Cost function is C = 35x + 500
    Demand Function is p = 50 - 0.1(sqrt x)

    Please include full solution. Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member DeMath's Avatar
    Joined
    Nov 2008
    From
    Moscow
    Posts
    473
    Thanks
    5
    Quote Originally Posted by meli3000 View Post
    find the price per unit p that produces the max profit P.

    Cost function is C = 35x + 500
    Demand Function is p = 50 - 0.1(sqrt x)

    Please include full solution. Thanks
    So, you have a Cost function C = 35x + 500 and a Demand function p = 50 - 0.1\sqrt x  = 50 - \frac{{\sqrt x }}{{10}}.

    Find a function of Total revenue (TR)

    TR = x \cdot p = x\left( {50 - \frac{{\sqrt x }}{{10}}} \right) = 50x - \frac{{{{\left( {\sqrt x } \right)}^3}}}{{10}}.

    Find a Profit-function (P)

    P = TR - C = 50x - \frac{{{{\left( {\sqrt x } \right)}^3}}}{{10}} - \left( {35x + 500} \right) = 15x - \frac{{{{\left( {\sqrt x } \right)}^3}}}{{10}} - 500.

    Now find the derivative of the Profit-function (P) and \frac{{dP}}{{dx}} = 0 and find x

    \frac{{dP}}{{dx}} = \frac{d}{{dx}}\left( {15x - \frac{{{{\left( {\sqrt x } \right)}^3}}}{{10}} - 500} \right) = 15 - \frac{{3\sqrt x }}{{20}} = 0 \Leftrightarrow \sqrt x  = 100 \Leftrightarrow \boxed{x = 10000}

    {P_{\max }} = P\left( {10000} \right) = 15 \cdot 10000 - \frac{{{{\left( {\sqrt {10000} } \right)}^3}}}{{10}} - 500 = 49500

    Finally, find the price per unit that produces the max profit

    p = p\left( {10000} \right) = 50 - 0.1\sqrt {10000}  = 50 - \frac{{\sqrt {10000} }}{{10}} = \boxed{40}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Maximum profit question
    Posted in the Calculus Forum
    Replies: 3
    Last Post: July 17th 2011, 09:17 AM
  2. Maximum profit.
    Posted in the Calculus Forum
    Replies: 5
    Last Post: November 3rd 2010, 10:27 AM
  3. Maximum Profit
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 15th 2010, 12:39 PM
  4. Calculating maximum profit
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: October 11th 2009, 06:24 PM
  5. Maximum profit questions
    Posted in the Calculus Forum
    Replies: 4
    Last Post: May 14th 2009, 06:44 PM

Search Tags


/mathhelpforum @mathhelpforum