Question

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) = 25x
c(x) = 12.5x + 50,000
(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 = \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.

Explanation:

Step1: Find break - even point (R = C)

Set \( R(x)=C(x) \), so \( 25x = 12.5x+50000 \). Subtract \( 12.5x \) from both sides: \( 25x - 12.5x=12.5x + 50000-12.5x \), which simplifies to \( 12.5x = 50000 \).

Step2: Solve for x

Divide both sides by \( 12.5 \): \( x=\frac{50000}{12.5}=4000 \).

Step3: Solve the inequality \( R(x)>C(x) \)

Substitute \( R(x) = 25x \) and \( C(x)=12.5x + 50000 \) into the inequality: \( 25x>12.5x + 50000 \). Subtract \( 12.5x \) from both sides: \( 25x-12.5x>12.5x + 50000 - 12.5x \), getting \( 12.5x>50000 \). Divide both sides by \( 12.5 \): \( x > 4000 \).

Answer:

(a) \( x = 4000 \)
(b) \( x>4000 \)