Sizu Riatto
Calculus · Chapter 14

Best Price Finder

Revenue is price × sales, and sales fall as the price rises. The best price is where the slope of profit is zero: where one more dime of price stops paying for the cups it loses.

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Opens on Chapter 14’s example: Hollis’s coffee: $3.50 a cup, 120 cups a morning, about 4 cups lost for each 10¢ added, 90¢ a cup in cost.

1. Your price and a straight demand line
2. How much to order at once
4. The cheapest speed for a delivery van
3. The cheapest average cost
Enter your numbers, then press Work it out.

How it works

A straight demand line through your price and sales, losing the same number of sales for each step of price, gives sales q = a − kp. Revenue p × q is a hill whose top is at p = a ÷ 2k. Profit is (p − cost) × q, a hill whose top is halfway between the revenue-best price and a k-step above it: in fact the profit-best price is the revenue-best price plus half the cost per unit. The demand line is a guess, so treat the answer as a starting point to test, not a law. For ordering, the yearly cost is (orders a year × cost per order) + (half an order in storage × storage cost); its slope is zero at √(2 × units × order cost ÷ storage cost).

From the book, Chapter 14. The line through $3.50 and 120 cups losing 40 cups a dollar is q = 260 − 40p. Revenue is best at $3.25 (130 cups, $422.50); profit at $3.70 (112 cups, $313.60, against $312.00 at $3.50), a shift of half the 90¢ cost. Ordering 2,400 bags a year at $50 an order and $2.40 a bag-year: 316 bags at a time, about $758.95 a year. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. Functions can be typed as 16t², 6 + 48t − 16t^2, 1000e^(0.05t) or √x; answers are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.