Based upon market research, the Hawthorne Company has determined that consumers

Question

Based upon market research, the Hawthorne Company has determined that consumers are willing to purchase 128 units of their portable media player each week when the price is set at $156.00 per unit. At a unit price of $60.00, consumers are willing to buy 320 units per week. (a) Determine the weekly demand equation for this product, assuming price, p, and quantity, x, are linearly related. (b) Determine the weekly revenue function. A(x) = (c) Determine the number of units consumers will demand weekly when the price is $116.00 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

Explanation / Answer

we have 2 points ( 128 , 156 ) and ( 320 , 60 )

finding equation from these two points

slope = ( 60 - 156 )/ ( 320 - 128 ) = -1/2

equation of price is

p = -1/2 x + 220

b) revenue = price * number of units

R(x) = (-1/2 x + 220 )x

R(x) = -1/2 x^2 + 220 x

c) units consumer will demand when price = 116

plug p = 116 and solve for x

116 = -1/2 x + 220

x = 208

consumer will demand 208 units

d) revenue is maximum at x = -220 / 2 ( -1/2 )

x = 220 units

e) price of each unit when revenue is maximized is

p = -1/2 (220) + 220 = 110 dollars

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