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 $6, and they sell an average of 20 per week at a price of $10 each. 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. 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? c) What is the maximum profit?

Answer 1

Answer:

N(x) = 40 - 2x

P(x) = -2x² + 52 x - 240

maximum profit = 13

Step-by-step explanation:

given data

feeder cost = $6

average sell = 20 per week

price = $10 each

solution

we consider here price per feeder = x

and profit per feeder  id here formula   = x - 6

so that here

total profit will be

P (x)  = ( x - 6 ) Nx

here N(x) is number of feeders sold at price =  x

so formula for N (x)  is here

N(x) = 20 - 2 ( x - 10 )    

N(x) = 40 - 2x

so that

P(x) = (x-6) ( 40 - 2x)

P(x) = -2x² + 52 x - 240

since here

a = -2

b = 52

c = -240

a < 0

so quadratic function have maximum value of c - $\frac{b^2}{4a}$

so it will be

maximum value = -240 - $\frac{52^2}{4(-2)}$

maximum value = 98

so here maximum profit attained at

x = $\frac{-b}{2a}$

x = $\frac{-52}{2(-2)}$

x = 13

maximum profit = 13