91Ó°ÊÓ

The point at which a company’s profits equal zero is called the company’s break-even point. Let R represent a company’s revenue, let C represent the company’s costs, and let x represent the number of units produced and sold each day.

$$\begin{gathered} R\left(x\right)=8x \\ C\left(x\right)=4.5x+17500 \end{gathered}$$

Find the firm’s break-even point.

So, Find x when $R=C$.

We have,

role="math" localid="1646808327530" $$\begin{gathered} R\left(x\right)=8x \\ C\left(x\right)=4.5x+17500 \end{gathered}$$

Equate the function R and C.

$$\begin{gathered} R\left(x\right)=C\left(x\right) \\ 8x=4.5x+17500 \\ 3.5x=17500 \\ x=5000 \end{gathered}$$

The firm’s break-even point is 5000.

Find the number of units that the company must sell to earn a profit.

x is the number of units that the company must sell to earn a profit when$R\left(x\right)>C\left(x\right)$

Find value of x for role="math" localid="1646808966512" $R\left(x\right)>C\left(x\right)$.

role="math" localid="1646809024502" $$\begin{gathered} R\left(x\right)>C\left(x\right) \\ 8x>4.5x+17500 \\ 3.5x>17500 \\ x>5000 \end{gathered}$$