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) =

Question

Mathematics

Abraham Fernandez

16 December, 03:23

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) =

+3

Answers (1)

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Stud · Answer 1

Step-by-step 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