MATH SOLVE

2 months ago

Q:
# 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?

Accepted Solution

A:

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