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 the society sells an average of 30 per week at a price of $10 each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses 3 sales per week.a) Find a function that models weekly profit in terms of price per feeder.b) What price should the society charge for each feeder to maximize profits? What is the maximum weekly profit?

Answer:The required function is: [tex]P(x)=-3x^2+78x-360[/tex]The price for each feeder to maximize profit should be $13.The maximum weekly profit is $147Step-by-step explanation:Consider the provided information.Part (A)The materials for each feeder cost $6, and the society sells an average of 30 per week at a price of $10 each.Let x be the price per feeder.Profit = Revenue - cost So the profit per feeder is [tex]x-6[/tex]If we increase then for every dollar increase, it loses 3 sales per week.This can be written as:[tex]30-3(x-10)[/tex][tex]30-3x+30[/tex][tex]-3x+60[/tex]Where x-10 represents the increase in price and 3  represents the decrease in sales per week.Thus the profit will be:[tex]P(x)=(x-6)(-3x+60)[/tex][tex]P(x)=-3x^2+78x-360[/tex]Hence, the required function is: [tex]P(x)=-3x^2+78x-360[/tex]Part (B) What price should the society charge for each feeder to maximize profits? What is the maximum weekly profit?The above function is a downward parabola so the maximum will occur at vertex.The x coordinate of the vertex of parabola is: [tex]x=\frac{-b}{2a}[/tex]Substitute a=-3, b=78 and c=-360 in above. [tex]x=\frac{-78}{2(-3)}=13[/tex]Hence, the price for each feeder to maximize profit should be $13.Now substitute the value of x in [tex]P(x)=-3x^2+78x-360[/tex][tex]P(x)=-3(13)^2+78(13)-360=$147[/tex]Hence, the maximum weekly profit is $147