Question

A bicycle manufacturer uses the given expression to model the monthly profit from sales of a new model of bicycle, where x is the selling price of one bicycle, in dollars. At what selling prices for the bicycle will the manufacturer make neither a profit nor a loss?

Answer 1

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

$f(x)= -x^{2}+300x-20,000$

we know that

The manufacturer will make no profit nor a loss when the profit is equal to zero

so

f(x)=0

$-x^{2}+300x-20,000=0$

Multiply by -1 both sides

$x^{2}-300x+20,000=0$

The formula to solve a quadratic equation of the form $ax^{2} +bx+c=0$ is equal to

$x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}$

in this problem we have

$a=1\\b=-300\\c=20,000$

substitute in the formula

$x=\frac{300(+/-)\sqrt{-300^{2}-4(1)(20,000)}}{2(1)}$

$x=\frac{300(+/-)\sqrt{10,000}}{2}$

$x=\frac{300(+/-)100}{2}$

$x=\frac{300(+)100}{2}=200$

$x=\frac{300(-)100}{2}=100$

therefore

The manufacturer will have no profit or loss when the selling price equals $200 or $100.