A community birdwatching society makes and sells simple bird

A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost $6, and they sell an average of 26 per week at a price of $13each. They have been considering raising the price, so they conduct a survey and find that for every dollar increase they lose 2 sales per week. Find a function that models weekly profit in terms of price per feeder. P(x) = ________ What price should the society charge for each feeder to maximize profits? What is the maximum profit?

Solution

Let x units of bird feeders

C(x) = 6x

Demand function : we have two points (26 , 13) and (24 , 14)

slope = ( 14-13)/(24-26) = -1/2

D(x) = -x/2 +c ; 13 = -26/2 +c

c = 26

D(x) = -x/2 +26

Revenue, R(x) = x( -x/2 +26) = -x^2/2 +26x

Profit : R(x) - C(x) = -x^2/2 +26x -6x = -x^2/2 +20x

P(x) =-x^2/2 +20x

for maximum profit : x = -b/2a = -(20/2*-1/2)

x = 20

P(x) = -20^2/2 +400 = $ 200