The demand function for a certain brand of CD is given by p = −0.01x^2 − 0.2x + 13 where p is the unit price in dollars and x is the quantity demanded each week

Question

The demand function for a certain brand of CD is given by
p = −0.01x^2 − 0.2x + 13
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 + 0.7x + 2
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.)

Answer 1

So the equilibrium price is when the supply and demand curves are equal which leads to

 

-.01x² - .2x +13 = .01x² + .7x + 2    bring everythng to one side to get

0 = .o2x² +.9x -11                           multiply by 100 to get rid of the decimals

0 = 2x² + 90x - 1100                        divide by 2 to get leading coefficient = 1 to make factoring easier

0 = x² + 45x - 550

0 = (x+55)(x - 10)

x=-55 reject or x=10

 

So when 10 units are produced, the supply and demand curves are equal....which both come out to 10 as well(they intersect at (10,10)

 

So we need the area between the demand curve( the upper curve) and the horizontal drawn through the equilibrium point which is y=10.

 

So we need t integrate from 0 to 10 (the x's).... ∫ -.01 x² - .2x + 13 - 10 dx

 

-.01/3 x³ - .2/2 x² + 3x    from 10 to 0

giving you -1000/300 - 10 + 30   - [0-0+0] = 16.6666 which rounds to 17 to the nearest dollar.

 

Hope this helped