A community bird-watching society makes and sells simple bird feeders to raise m

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 $5, and the society sells an average of 40 per week at a price of $9 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 be the price per feeder and P(x) be the profit.)

Explanation / Answer

let $ p be the price increase per feeder. price per feeder will be= x = 9 + p dollars loss of sale /week will be= 2p feeders. and sale per week will be = (40 - 2p) feeders profit per week = P(x) = feeders sold / week * (price per feeder - cost per feeder) => P(x) = (40 - 2p) * ((9 + p) - 5) = (40 - 2p) * (4 + p) = 160 - 8p + 40p - 2p^2 = 160 + 32p - 2p^2 This is the required function. It is quadratic function. for p = 0, P(x) = 160 for p = 1, P(x) = 190 for p = 2, P(x) = 216 ...... for p = 15, P(X) = 30 and so on it can be computed. maxima occurs at p = -b/2a = -32/-2*2 = 8 and P(X)max = 288

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