Question 1178251

 For a certain company, the cost function for producing {{{x }}}items is

 {{{C(x)=50x+200 }}}

and the revenue function for selling {{{x}}} items is 

{{{R(x)=-0.5(x-120)^2+7200}}}

The maximum capacity of the company is {{{140}}} items.

Assuming that the company sells all that it produces, what is the profit function?

{{{P(x)=R(x)- C(x)}}}

{{{P(x)=-0.5(x-120)^2+7200-(50x+200 )}}}


What is the domain of {{{P(x)}}}?

 Note that the company can sell at least{{{ 0}}} items and at most {{{140 }}}items. Thus, the domain is

{{{0<=x<=140}}}

The company can choose to produce either {{{70}}} or {{{80}}} items. What is their profit for each case, and which level of production should they choose?

Profit when producing {{{70}}} items =

so, if {{{x=70}}}

{{{P(x)=-0.5(70-120)^2+7200-(50*70+200 )}}}
{{{P(x)=5950 - 3700}}}
{{{P(x)=2250}}}


Profit when producing {{{80}}} items =

if {{{x=80}}}

{{{P(x)=-0.5(80-120)^2+7200-(50*80+200 )}}}

{{{P(x)=6400 - 4200}}}

{{{P(x)=2200}}}

Can you explain, from our model, why the company makes less profit when producing 10 more units? 

there is max number of units where square function (parabola) goes up, after that goes down 

the maximum is when {{{x=-b/2a}}}

{{{P(x)=-0.5(x-120)^2+7200-(50x+200 )}}}-expand

{{{P(x) = -0.5x^2 + 70 x - 200}}}

The maximum is: {{{x=-b/2a=-70/(2*-0.5)=-70/-1=70}}} or {{{70 }}} units 

 => for {{{80}}} units the {{{cost}}} is {{{higher}}} and profit lower