Quadratic Functions

Question

a lawn care company charges $20 a week and has 120 clients. The owner estimates that for each $2 increase in its fee, the company would lose 5 clients. What fee should the owner charge to receive the maximum revenue?

Answer 1

Revenue = Price times Quantity, R=PQ

The company is currently making 20(120) = $2400 per week

Price = 20+2x

Quantity = 120-5x

R = (20 + 2x)(120-5x) = 2400 -100x +240x -10x² = -10x² + 140x +2400 = ax²+bx+c

This is a parabola opening down. The vertex is at, x = -b/2a = -140/(-20) = 7

The owner should charge (20+2(7)) = $34

But they will only have 120-5(7) = 85 clients

So the maximum revenue will be 34(85) = $2890