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. The revenue function is given by r(x) = 20x and the cost function is given by c(x) = 12.5x + 37,500. (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.

Answer 1

The firm's break-even point is when it sells 3,000 units, as this is where revenue equals cost. To earn a profit, the firm must sell more than 3,000 units per day.

Step-by-step explanation:

Break-Even Point and Profit Analysis

To find the firm's break-even point, we set the revenue function equal to the cost function and solve for x. Given the revenue function is r(x) = 20x, and the cost function is c(x) = 12.5x + 37,500, the break-even point is where:

20x = 12.5x + 37,500

Solving for x gives:

x = 3,000

This means that the firm must sell 3,000 units to break-even. To determine the values of x for which the firm makes a profit (r(x) > c(x)), we look for the number of units such that:

20x > 12.5x + 37,500

Solving the inequality, we find:

x > 3,000

Therefore, the company must sell more than 3,000 units per day to earn a profit.