Linear Cost, Revenue & Profit

Business Calculus · Axiom Academy

LESSON Linear Cost, Revenue & Profit The fundamental business functions and their linear relationships A company's total cost to produce goods typically has two components: Costs that don't change with production level Cost that increases with each unit A t-shirt company has fixed costs of 2,000/month (rent, equipment) and spends 8 per shirt on materials and labor. To produce 100 shirts: C(100) = 2000 + 8(100) = \ 2,800 Revenue is the total income from selling goods. In the simplest case, with a fixed selling price: where p = price per unit and x = quantity sold When price is constant, revenue is simply price times quantity—a linear function through the origin. Later, we'll see models where price depends on quantity (demand curves), leading to non-linear revenue. The company sells each t-shirt for 20. Selling 100 shirts: R(100) = 20(100) = \ 2,000 Profit is what remains after subtracting costs from revenue: For linear cost and revenue functions: Profit contribution per unit sold The "hole" profit must climb out of P(x) = 20x - (2000 + 8x) = 12x - 2000 At 100 shirts: P(100) = 12(100) - 2000 = -\ 800 (a loss!) Each shirt contributes 12 toward fixed costs and profit. Visualizing Cost, Revenue & Profit Let's see all three functions together for our t-shirt business: Where Revenue equals Cost, Profit is zero: The company must sell at least 167 shirts to break even! In linear functions, the slope tells us the rate of change—a preview of what derivatives will formalize:

This is the written version of the interactive lesson above. See the full Business Calculus course.