The management of the Titan Tire Company has determined that the quantity demanded x of their Super Titan tires/week is related to the unit price p by the relation

Question

p = 160 − x^2
where p is measured in dollars and x is measured in units of a thousand. Titan will make x units of the tires available in the market if the unit price is
p = 64 + 1/2(x^2)
dollars. Determine the consumers' surplus and the producers' surplus when the market unit price is set at the equilibrium price. (Round your answers to the nearest dollar.)
consumer's surplus = ?
producer's surplus = ?

Answer 1

64 + 1/2x^2 = 160 − x^2

3*x^2/2 -96 = 0

3x^2/2 -96=0

3/2(x^2-64) = 0

3/2(x-8)(x+ 8) = 0

x = 8
x = -8

Solving gives x=8
This is the equilibrium quantity
Plug 8 into either demand or supply function to get equilibrium price = 96

Consumer surplus:
ʃ (160-x^2 )dx on [0,8] - 8*96

(160x -x^3/3 on [0,8] - 768

1280-512/3 - 768 = 341.33

Producer surplus:
8*96 -ʃ (64 1x^2/2 )dx on [0,8]

768 - ((64x + x^3/6)) on [0,8]

768 - (512 + 256/3)

768 -1792/3 = 170.67