Q. 43 - 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 .

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

Step-by-Step Solution

Answer

Part (a) $x=5000$. The firm’s break-even point is 5000. 

Part (b) $x>5000$

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.

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

Part (a). Step 2. Explanation.

Find the firm’s break-even point. 

So, Find x when $R=C$.

We have,

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

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.

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

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