Differential word problem

Question

When a wholesaler sold a product at $50 per unit, sales were 184 units per week. After a price increase of $5, however, the average number of units sold dropped to 164 per week. Assuming that the demand function is linear, what price per unit will yield a maximum total revenue?

Answer 1

A linear demand means that the price P and the demand D (number of units sold) form a line, which would have an equation like this, with unknown (currently) coefficients b and c:

P + b D = c or D = (c - P)/b

We have two data points, with (P, D) values (50,184) and (55,164), and two unknown coefficients, so we can solve to find them:

50 + 184 b = c

55 + 164 b = c

Subtracting the two equations, we have -5 + 20 b = 0, or b = 0.25. Substituting back in, we get c = 96.

We're trying to find the price P that will maximize the revenue, which is P times D, or P · (c - P) / b, or (multiplying this out): - P² / b + P c / b

The maximum revenue will occur at the spot where the derivative of this is zero, so:

0 = - 2 P / b + c/b

giving

P = c / 2 = 48.

So, the optimal price is 48, and at that price, we would expect to sell 192 units.