Based upon market research the Hawthorne Company has determi

Based upon market research, the Hawthorne Company has determined that consumers are willing to purchase 146 units of their portable media player each week when the price is set at $60.40 per unit. At a unit price of $12.60, consumers are willing to buy 385 units per week.

(a) Determine the weekly demand equation for this product, assuming price, p, and quantity, x, are linearly related.

p =

(b) Determine the weekly revenue function.
R(x) =  
(c) Determine the number of units consumers will demand weekly when the price is $35.60 per portable media player.
units
(d) Determine the number of units consumers will demand weekly when the revenue is maximized.
units
(e) Determine the price of each unit when the revenue is maximized.
dollars

Solution

146 units of their portable media player each week when the price is set at $60.40 per unit. At a unit price of $12.60, consumers are willing to buy 385 units per week

146 units ----- $60.40

385 units ----- $ 12.60

Using these two pointswe can find the demand equation : P(x) = mx +c

slope m = ( 385 -146)/ (12.60 -60.40) = -239/47.8 = -5

Find c: 385 = -5*12.60 +c----> c = 448

a) P = -5x +448

b) Revenue function : R(x) = x*P(x) = -5x^2 + 448x

c)P = $35.60

solve for x: 35.60 = -5x +448

x = 82 units

d) Revenue is maximised when dR/dx =0

-10x + 448=0

x = 44.8 = 45 units ( approx)

e) P(x) = 44.8

44.8 = -5x + 448

x= $80.64