Chapter 3.1, Problem 43AYU

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.

$R\left(x\right)=8x$

$C\left(x\right)=4.5x+17,500$

To determine

To calculate:

a. Find the breakeven point.

b. The value of $x$ when $R(x)>C(x)$.

Answer to Problem 43AYU

Solution:

a. $x=5000$

b. $x>5000$

Explanation of Solution

Given:

$R(x)=8x$

$C(x)=4.5x+17500$

Calculation:

a. Now, we need to find $x$ when $R(x)=C(x)$.

$R(x)=C(x)$

$\Rightarrow 8x=4.5x+17500$

$\Rightarrow 8x-4.5x=17500$

$\Rightarrow x=\frac{17500}{3.5}=5000$

The breakeven point of the company is at $x=5000$.

b. When $R(x)>C(x)$, we get

$R(x)>C(x)$

$\Rightarrow 8x>4.5x+17500$

$\Rightarrow 8x-4.5x>17500$

$\Rightarrow x>\frac{17500}{3.5}$

$\Rightarrow x>5000$