Question 1204775


Demand Function 

{{{p(x) = 80 - 0.1sqrt(x) }}}

Cost Function

{{{C(x) = 35x + 600}}}


To find the price that will maximize profit, using demand and cost functions, first step is to find the profit function {{{P(x)}}}. Then, use the first derivative to find the critical points for the profit function. Finally use the critical points to find the price to maximize profit.


Profit Function:

{{{P(x) ) = R(x) - C(x) }}}...revenue {{{R(x) =x*p(x)}}}

  {{{P(x) ) = x*p(x) - C(x) }}}

 {{{ P(x) ) = x*(80 - 0.1sqrt(x) ) - (35x + 600)}}}
 
 {{{P(x) = 80x - 0.1x^(3/2) - 35x -600 }}}


Then find first derivate and set equal to zero:


{{{(d/dx)(80x - 0.1x^(3/2) - 35x -600 )=-0.15 (sqrt(x) - 300)}}}

{{{-0.15 (sqrt(x) - 300)=0}}}

{{{sqrt(x) - 300=0}}}

{{{sqrt(x) =300}}}

{{{x=300^2}}}

{{{x = 90000}}}


 The maximum profit is {{{P(90000)}}}'.