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 = 172 − 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 = 76 + (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 _________

To find the consumers' and producers' surplus, we first need to determine the equilibrium price and quantity.

Equilibrium occurs when the quantity demanded equals the quantity supplied. In this case, the quantity demanded is given by p = 172 − x^2, and the quantity supplied is given by p = 76 + (1/2)x^2.

Setting these two equations equal to each other, we get:
172 − x^2 = 76 + (1/2)x^2

Combining like terms, we have:
2x^2 + x^2 = 172 - 76
3x^2 = 96
x^2 = 32
x = √32
x = 5.66 (rounded to two decimal places)

Now, substituting the value of x into either equation gives us the equilibrium price:
p = 172 - (5.66)^2
p = 172 - 32.11
p = 139.89 (rounded to two decimal places)

The equilibrium price is approximately $139.89, and the equilibrium quantity is approximately 5,660 units.

To find the consumers' surplus, we need to calculate the area under the demand curve but above the equilibrium price.

Consumers' Surplus:
To find the area under the demand curve, we integrate the demand function with respect to x from 0 to the equilibrium quantity:

CS = ∫[p(x)dx] from 0 to 5.66

Integrating p(x) = 172 - x^2 from 0 to 5.66:
CS = ∫[(172 - x^2)dx] from 0 to 5.66
CS = [172x - (1/3)x^3] from 0 to 5.66
CS = [(172 * 5.66) - (1/3)(5.66)^3] - [(172 * 0) - (1/3)(0)^3]
CS ≈ 487.91 (rounded to the nearest dollar)

The consumer's surplus is approximately $487.91.

Producer's Surplus:
To find the producer's surplus, we need to calculate the area above the equilibrium price but below the supply curve.

Producer's Surplus:
To find the area above the supply curve, we integrate the supply function with respect to x from 0 to the equilibrium quantity:

PS = ∫[p(x)dx] from 0 to 5.66

Integrating p(x) = 76 + (1/2)x^2 from 0 to 5.66:
PS = ∫[(76 + (1/2)x^2)dx] from 0 to 5.66
PS = [76x + (1/6)x^3] from 0 to 5.66
PS = [(76 * 5.66) + (1/6)(5.66)^3] - [(76 * 0) + (1/6)(0)^3]
PS ≈ 431.59 (rounded to the nearest dollar)

The producer's surplus is approximately $431.59.

In summary, the consumer's surplus is approximately $487.91, and the producer's surplus is approximately $431.59.

To determine the consumer's surplus and producer's surplus at the equilibrium price, we first need to find the equilibrium price and quantity.

At equilibrium, the quantity demanded (x) is equal to the quantity supplied (x). Therefore, we can set the two price equations equal to each other:

172 - x^2 = 76 + (1/2)x^2

Now we can solve for x:

172 - x^2 = 76 + (1/2)x^2
Multiply both sides by 2 to eliminate the fraction:
344 - 2x^2 = 152 + x^2
Rearrange the equation:
3x^2 = 192
Divide by 3:
x^2 = 64
Take the square root of both sides:
x = 8

Now we have the equilibrium quantity x = 8. To find the equilibrium price, we substitute this value back into one of the price equations:

p = 76 + (1/2)x^2
p = 76 + (1/2)(8^2)
p = 76 + 32
p = 108

The equilibrium price is $108 per unit.

To calculate the consumer's surplus, we need to integrate the demand curve from 0 to the equilibrium quantity (8) and subtract the area under the demand curve above the equilibrium price:

Consumer's surplus = ∫[0 to 8] (172 - x^2) dx - (108 * 8)
Let's calculate this:

Consumer's surplus = ∫[0 to 8] (172 - x^2) dx - (108 * 8)
= [172x - (1/3)x^3] from 0 to 8 - 864
= (172(8) - (1/3)(8^3)) - 864
= (1376 - (1/3)(512)) - 864
= (1376 - 170.67) - 864
= 1205.33 - 864
≈ 341 (rounded to the nearest dollar)

The consumer's surplus is approximately $341.

To calculate the producer's surplus, we need to integrate the supply curve from 0 to the equilibrium quantity (8) and subtract the area above the equilibrium price:

Producer's surplus = (108 * 8) - ∫[0 to 8] (76 + (1/2)x^2) dx
Let's calculate this:

Producer's surplus = (108 * 8) - ∫[0 to 8] (76 + (1/2)x^2) dx
= 864 - [76x + (1/6)x^3] from 0 to 8
= 864 - (76(8) + (1/6)(8^3)) - 0
= 864 - (608 + (1/6)(512))
= 864 - (608 + 85.33)
= 864 - 693.33
≈ 170 (rounded to the nearest dollar)

The producer's surplus is approximately $170.