Question 1135433

The cost, in dollars, for a company to produce x widgets is given by
{{{C(x) = 4050 + 9.00x}}} for  {{{x >=  0}}}, and 

the price-demand function, in dollars per widget, is
 
{{{p(x) = 63-0.03x}}} for{{{ 0 <= x <=2100}}}

In Quiz 2, problem #10, it was seen that the profit function for this scenario is

{{{P(x) = -0.03x^2 + 54.00x-4050}}}

(a) The profit function is a quadratic function and so its graph is a parabola. Does the parabola open up or down? ___{{{down}}}_______ 

(b) Find the vertex of the profit function P(x) using algebra. 

{{{P(x) = -0.03(x^2 -54.00x/0.03)-4050}}}
{{{P(x) = -0.03(x^2 -1800x)-4050}}}...complete square
{{{P(x) = -0.03(x^2 -1800x+b^2)-(-(0.03)(b^2))-4050}}}

recall:  {{{2ab=1800}}} 
since {{{a=1}}}, we have {{{2b=1800}}}...solve for {{{b}}}

 {{{b=900}}} 


then we have

{{{P(x) =  - 0.03 (x^2 -1800x+900^2)-(-(0.03)(900^2))-4050}}}

{{{P(x) =  - 0.03 (x -900)^2)+24300-4050}}}

 {{{P(x) =  - 0.03 (x - 900)^2+20250}}}

=>{{{h=900}}} and {{{k=20250}}}
vertex is at: ({{{900}}},{{{20250}}})


(c) State the maximum profit and the number of widgets which yield that maximum profit: 
The maximum profit is _____{{{20250}}} ________, when ___{{{x=900}}}________ widgets are produced and sold. 

(d) Determine the price to charge per widget in order to maximize profit. 

{{{p(x) = 63-0.03x}}}...plug in {{{x=900}}}

{{{p(x) = 63-0.03*900}}}
{{{p(x) = 63-27}}}

{{{p(x) = 36}}}


(e) Find and interpret the break-even points. Show algebraic work. 

Break-Even _Point => when {{{C(x) =P(x)}}}

if
{{{C(x) = 4050 + 9.00x}}}
{{{P(x) = -0.03x^2 + 54.00x-4050}}}

then {{{4050 + 9.00x=-0.03x^2 + 54.00x-4050}}}

{{{4050 + 9.00x+0.03x^2 - 54.00x+4050=0}}}

{{{0.03 x^2 - 45 x + 8100 = 0 }}}...using quadratic formula we get:

{{{x}}}≈{{{209.167}}}
{{{x}}}≈{{{1290.83}}}