Answer:
The question have no a clear question to resolved however is common in this kind of exercises to know
a). Profit function [tex]P(x)= 1824x-x^{2} -43200[/tex]
b). Break points [tex]x_{1} =24 , x_{2} =1800[/tex]
c). Maximum value in revenue function [tex]R(x)= 209383,05[/tex]
Step-by-step explanation:
a). Cost per Unit = [tex]\frac{4}{9}*x + 333[/tex]
Variable Cost = [tex](\frac{4}{9}*x + 333) * x[/tex]
Fixed Cost = 43200 $
Total cost = Fixed Cost + Variable Cost ⇒[tex]C(x)= 43200 + \frac{4}{9}*x^{2} + 333 x[/tex]
Given selling price per Units = [tex]2157 - \frac{5}{9}*x[/tex]
Then selling price for 'x' units is Revenue Function = [tex](2157 - \frac{5}{9}*x)* x[/tex]
Profit function can be find by Revenue - Total Cost:
[tex]P(x)= R(x)-C(x)[/tex]
[tex]P(x)= (2157 x -\frac{5}{9} *x^{2} ) - (\frac{4}{9}*x^{2} + 333x + 43200)[/tex]
[tex]P(x)= 1824x - x^{2} -43200[/tex]
b). The break points is the total cost equal to selling cost
[tex](\frac{4}{9}*x^{2} + 333* x + 43200 = 2157x - \frac{5}{9}*x^{2}[/tex]
[tex]x^{2} -1824x +43200=0[/tex]
Using this equation to know the X value:
[tex]\frac{-b+/- \sqrt{b^{2}- 4*a*c } }{2*a}[/tex]
[tex]\frac{-(-1824+/-\sqrt{1824^{2} -4*43200} }{2}[/tex]
[tex]912 +/- 888[/tex]
[tex]x_{1}=24 ,x_{2}=1800[/tex]
So, it will take 24 or 1800 to break even points
c). Substitute [tex]2157-\frac{5}{9}*x[/tex] for a vertex formula [tex]x=-\frac{b}{2*a}[/tex]
[tex]x=-\frac{2157}{2*\frac{5}{9}}[/tex]
[tex]x=1941,3[/tex]
Now substitute 1941,3 for x in revenue function
[tex]R(1941,3)= 2157 (1941,3)-\frac{5}{9}*1941,3^{2}[/tex]
[tex]R(1941,3)=2093692,05[/tex]
