!math (4)
posted by Mia on .
The demand function for a certain brand of CD is given by
p = −0.01x^2 − 0.2x + 12
where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. The supply function is given by
p = 0.01x^2 + 0.5x + 3
where p is the unit price in dollars and x stands for the quantity that will be made available in the market by the supplier, measured in units of a thousand. Determine the producers' surplus if the market price is set at the equilibrium price. (Round your answer to the nearest dollar.)

0.01x^2 + 0.5x + 3 = −0.01x^2 − 0.2x + 12
0.02^2 + 7x 9 = 0
2x^2 + 70x 900 =0
2(x^2 + 35x 450) = 0
2(x 10)(x + 45)=0
x = 10
x = 45
Solving gives x=10
This is the equilibrium quantity
Plug 10 into either demand or supply function to get equilibrium price = 9
Consumer surplus:
ʃ (−0.01x^2 − 0.2x + 12 )dx on [0,10]  10*9
(.01/3x^3/3 .2x^2/3 + 12x on [0,10]  90
10/3  20/3 + 120 90 = 20
Producer surplus:
10*9 ʃ (0.01x^2 + 0.5x + 3)dx on [0,10]
90  ((.01/3)x^3 + .5x^2/2 + 3x)) on [0,10]
90  ( 10/3 + 25+ 30)
90 175/3 = 31.67