Based upon market research the Hawthrone company has that co

Based upon market research, the Hawthrone company has that consumers are willing to purchase 150 units of their portable media player each week when the price is set at $20,60 per unit. At a unit price of $5.10, consumers are willing to by 305 units per week. Determine the weekly demand equation for this product, assuming price, p, and quantity, x, are linearly related. Determine the weekly revence function. Determine the number of units consumers will demand weekly when price is $18,80 per portable media plkayer. Determine the number of units consumers will demand weekly when the revenue is maximized. Determine the price of each unit when the revenue is maximized.

Solution

Points ---( 150, $20.60,) and ( 305, $5.10)

Find dlope from these two points :( 5.10 -20.60)/(305-150) =-15.5/155 = -0.1

Let x be the qty and p is the price per unit.

p = mx + b . we need to find b form one of the points ---( 150, $ 20.60)

p = -0.1x +b ----> 20.60 = -0.1*150 +b----> b= 35.6

a) So, p = -0.1x + 35.6

b) Revenue function R(x) = p*x = x( -0.1x + 35.6) = -0.1x^2 +35.6x

c) p = $ 18.80 Find x:

18.80 = -0.1x + 35.6

x = 168 units per week

d) R(x) is maximised at the point dR/dx =0

R(x) =  -0.1x^2 +35.6x

dR/dx = -0.2x + 35.6 =0

-0.2x + 35.6 =0----> x = 178 units

find p(x) for x = 178 units :p= -01*178 + 35.6 = $ 17.8