Based upon market research the Hawthorne Company has determi

Based upon market research, the Hawthorne Company has determined that consumers are willing to purchase 141 units of their portable media player each week when the price is set at $35.10 per unit. At a unit price of $12.20, consumers are willing to buy 370 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 $28.00 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

At $35.10 ---- 141 units

At $ 12.20 ---- 370 units

Wefind the demand function from these two points the way we find the equation of linear function

P(x) =mx+c

where slope m = (370 -141)/( 12.20 - 35.10) = -10

P = -10x +c .Find c

141 = -10*35.10 +c ----> c= 492

a)   So, P(x) = -10x + 492

b)   Revenue R(x) = x* P(x) = x( -10x +492) = -10x^2 +492x

c)  x = $ 28.00. Find x from demand function

Find P = -10*28 + 492 = 212 units

d) number of units consumers will demand weekly when the revenue is maximized. units

Find dR/dx = -20x +492

20x = 492 ---> x= 492/20 = $24.5

P( x= 24.5) = -10*24.5 +492 = 247 units