Question

A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost $3, and the society sells an average of 28 per week at a price of $7 each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses 2 sales per week.

(a) Find a function that models weekly profit in terms of price per feeder. (Let x represent the price per feeder and P represent the profit.)
P(x) =

Answer 1

Explanation:

Let x = number of $1 increases

Profit = Revenue - cost

Revenue = (price)(quantity) = (x+7)(28-2x)

Cost = (cost per feeder)(number of feeders) = 3(28-2x)

P(x) = profit = (x+7)(28-2x) - 3(28-2x)

= (28-2x)(x+4)

= -2x2 + 28x + 144

The graph of the profit function is a parabola opening downward. The profit is maximized when x is the x-coordinate of the vertex.

Maximum profit when x = -28/(2(-2)) = 7

To maximize profit, increase the price by $7 per feeder. So, the price per feeder should be $14.

Maximum profit = P(7) = -2(7)2 + 28(7) + 144 = $242/week