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

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 = ?

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

To determine the consumer's surplus and the producer's surplus, we need to first find the equilibrium price, which occurs when the quantity demanded equals the quantity supplied.

In this case, the quantity demanded is given by p = 160 - x^2, and the quantity supplied is given by p = 64 + 1/2(x^2). Setting these two equations equal to each other will give us the equilibrium price and quantity.

160 - x^2 = 64 + 1/2(x^2)

To solve this equation, we can combine like terms:

1/2(x^2) + x^2 = 160 - 64

Simplifying further:

3/2(x^2) = 96

Next, we multiply both sides by 2/3 to isolate x^2:

x^2 = (96 * 2/3)

x^2 = 64

To find x, we take the square root of both sides:

x = √64

x = 8

Now, we know that at the equilibrium price, x = 8 units of tires will be supplied and demanded.

To find the equilibrium price, we substitute this value of x into either of the given price equations. Let's use p = 64 + 1/2(x^2):

p = 64 + (1/2)(8^2)

p = 64 + 32

p = 96

Therefore, the equilibrium price is $96.

Now, we can calculate the consumer's surplus and the producer's surplus.

The consumer's surplus is the difference between what consumers are willing to pay (based on their demand) and what they actually pay at the equilibrium price.

To calculate the consumer's surplus, we need to integrate the demand equation from 0 to 8 (the equilibrium quantity) and subtract it from the area under the equilibrium price line (p = 96) up to x = 8.

The demand equation is given by p = 160 - x^2. Integrating this equation from 0 to 8:

∫[0 to 8] (160 - x^2) dx

= [160x - (1/3)x^3] evaluated from 0 to 8

= [160(8) - (1/3)(8^3)] - [160(0) - (1/3)(0^3)]

= 1280 - 213.33

= 1066.67

The area under the equilibrium price line (p = 96) up to x = 8 is:

∫[0 to 8] 96 dx

= 96x evaluated from 0 to 8

= 96(8) - 96(0)

= 768

Therefore, the consumer's surplus is:

consumer's surplus = 768 - 1066.67

consumer's surplus ≈ -298.67

Since the consumer's surplus cannot be negative, we round it to zero for practical purposes. The consumer's surplus is approximately $0.

The producer's surplus is the difference between the revenue producers receive at the equilibrium price and the minimum amount they are willing to accept.

To calculate the producer's surplus, we need to integrate the supply equation from 0 to 8 (the equilibrium quantity) and subtract it from the area under the equilibrium price line (p = 96) up to x = 8.

The supply equation is given by p = 64 + 1/2(x^2). Integrating this equation from 0 to 8:

∫[0 to 8] (64 + 1/2(x^2)) dx

= [64x + (1/6)x^3] evaluated from 0 to 8

= [64(8) + (1/6)(8^3)] - [64(0) +(1/6)(0^3)]

= 512 + 85.333

= 597.333

Therefore, the producer's surplus is:

producer's surplus = 597.333 - 768

producer's surplus ≈ -170.667

Again, since the producer's surplus cannot be negative, we round it to zero for practical purposes. The producer's surplus is approximately $0.

In conclusion:
Consumer's surplus ≈ $0
Producer's surplus ≈ $0