Q. 44 - Precalculus Enhanced with Graphing Utilities

Question

The point at which a company’s profits equal zero is called the company’s break-even point. For Problems 43 and 44, 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. 

(a) Find the firm’s break-even point; that is, find x so that $R=C$.

(b) Find the values of x such that $R\left(x\right)>C\left(x\right)$. This represents the number of units that the company must sell to earn a profit.

$$\begin{gathered} R\left(x\right)=12x \\ C\left(x\right)=10x+15000 \end{gathered}$$.

Step-by-Step Solution

Answer

Part (a) $x=7500$. The firm’s break-even point is 7500.

Part (b) $x>7500$.

Part (a) Step 1. Given Information.

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.

Part (a). Step 2. Explanation.

Find the firm’s break-even point.

So, Find x when $R=C$

We have ,

$$\begin{gathered} R\left(x\right)=12x \\ C\left(x\right)=10x+15000 \end{gathered}$$

Equate R and C.

$$\begin{gathered} R\left(x\right)=C\left(x\right) \\ 12x=10x+15000 \\ 2x=15000 \\ x=7500 \end{gathered}$$

The firm’s break-even point is 7500.

Part (b). Step 1. Explanation.

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 $R\left(x\right)>C\left(x\right)$.

$$\begin{gathered} R\left(x\right)>C\left(x\right) \\ 12x>10x+15000 \\ 2x>15000 \\ x>7500 \end{gathered}$$