A community bird-watching society

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 36 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.)

 

(b) What price should the society charge for each feeder to maximize profits?

 

(c) What is the maximum weekly profit? 

Answer 1

Let x = number of $1 increases

 

Profit = Revenue - cost 

 

Revenue = (price)(quantity) = (x+7)(36-2x)

 

Cost = (cost per feeder)(number of feeders) = 3(36-2x)

 

P(x) = profit = (x+7)(36-2x) - 3(36-2x)

 

                  = (36-2x)(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