Answer:
The manufacturer will have no profit or loss when the selling price equals $200 or $100.
Step-by-step explanation:
The expression to model the monthly profit from sales of a new model of bicycle is
-x^2 + 300x -20,000
Let
f(x) -----> the monthly profit in dollars
x -----> s the selling price of one bicycle in dollars
[tex]f(x)= -x^{2}+300x-20,000[/tex]
we know that
The manufacturer will make no profit nor a loss when the profit is equal to zero
so
f(x)=0
[tex]-x^{2}+300x-20,000=0[/tex]
Multiply by -1 both sides
[tex]x^{2}-300x+20,000=0[/tex]
The formula to solve a quadratic equation of the form [tex]ax^{2} +bx+c=0[/tex] is equal to
[tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex]
in this problem we have
[tex]a=1\\b=-300\\c=20,000[/tex]
substitute in the formula
[tex]x=\frac{300(+/-)\sqrt{-300^{2}-4(1)(20,000)}}{2(1)}[/tex]
[tex]x=\frac{300(+/-)\sqrt{10,000}}{2}[/tex]
[tex]x=\frac{300(+/-)100}{2}[/tex]
[tex]x=\frac{300(+)100}{2}=200[/tex]
[tex]x=\frac{300(-)100}{2}=100[/tex]
therefore
The manufacturer will have no profit or loss when the selling price equals $200 or $100.
