Questions: 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. R(x)=12x C(x)=6.5x+22,000 (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(x)>C(x). This represents the number of units that the company must sell to earn a profit. (a) x= (Type a whole number.) (b) Solve the inequality for x. Type the correct inequality symbol in the first answer box below, and type an integer in the second answer box. x

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.

R(x)=12x
C(x)=6.5x+22,000

(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(x)>C(x). This represents the number of units that the company must sell to earn a profit.
(a) x= (Type a whole number.)
(b) Solve the inequality for x. Type the correct inequality symbol in the first answer box below, and type an integer in the second answer box.
x

Transcript text: 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. \[ \begin{array}{l} R(x)=12 x \\ C(x)=6.5 x+22,000 \end{array} \] (a) Find the firm's break-even point; that is, find $x$ so that $R=C$. (b) Find the values of $x$ such that $R(x)>C(x)$. This represents the number of units that the company must sell to earn a profit. (a) $x=$ $\square$ (Type a whole number.) (b) Solve the inequality for $x$. Type the correct inequality symbol in the first answer box below, and type an integer in the second answer box. $x$ $\square$

Solution

Solution Steps

Step 1: Find the Break-Even Point

To find the break-even point, we set the revenue function equal to the cost function. Thus, we solve the equation $R(x) = C(x)$, which gives us $a x = b x + c$.

Step 2: Solve for x

Rearranging the equation gives us $x = \frac{c}{a - b}$. Substituting the given values, we find $x = 4000$.

Step 3: Find the Range of Values for x for Which R(x) > C(x)

To find this range, we solve the inequality $a x > b x + c$. Simplifying the inequality gives us $x > \frac{{c}}{{a - b}}$, indicating the company must sell more than this number of units to earn a profit.