Find the consumers' surplus and producers' surplus for

p = D(x) = 71 - (1/10)x and
p = S(x) = 35 + (1/20)x

To find the consumer's surplus, we first need to find the equilibrium price and quantity. The equilibrium occurs when demand (D(x)) equals supply (S(x)).

Setting D(x) equal to S(x), we have:

71 - (1/10)x = 35 + (1/20)x

To solve for x, let's isolate the variables:

71 - 35 = (1/20)x + (1/10)x

36 = (3/20)x

Multiplying both sides by 20/3:

(20/3) * 36 = x

240 = x

So, the equilibrium quantity is x = 240.

To find the equilibrium price, we substitute x = 240 into either D(x) or S(x). Let's use S(x):

p = S(240) = 35 + (1/20)(240) = 35 + 12 = 47

Therefore, the equilibrium price is p = 47.

Consumers' Surplus:

Consumer's surplus is the difference between the maximum price consumers are willing to pay and the actual price they pay. It can be calculated by finding the area of the triangle below the demand curve and above the equilibrium price.

The maximum price consumers are willing to pay can be found by substituting x = 0 into the demand equation:

p = D(0) = 71 - (1/10)(0) = 71

So, the maximum price consumers are willing to pay is 71.

The consumer's surplus can be calculated using the formula for the area of a triangle:

Consumer's Surplus = 1/2 * (Base) * (Height)

In this case, the base is the quantity (240) and the height is the difference between the maximum price (71) and the equilibrium price (47):

Consumer's Surplus = 1/2 * (240) * (71 - 47)

Consumer's Surplus = 1/2 * (240) * (24)

Consumer's Surplus = 2880

Therefore, the consumer's surplus is 2880.

Producers' Surplus:

Producer's surplus is the difference between the minimum price producers are willing to receive and the actual price they receive. It can be calculated by finding the area of the triangle above the supply curve and below the equilibrium price.

The minimum price producers are willing to receive can be found by substituting x = 0 into the supply equation:

p = S(0) = 35 + (1/20)(0) = 35

So, the minimum price producers are willing to receive is 35.

The producer's surplus can be calculated using the same formula for the area of a triangle:

Producer's Surplus = 1/2 * (Base) * (Height)

In this case, the base is the quantity (240) and the height is the difference between the equilibrium price (47) and the minimum price (35):

Producer's Surplus = 1/2 * (240) * (47 - 35)

Producer's Surplus = 1/2 * (240) * (12)

Producer's Surplus = 1440

Therefore, the producer's surplus is 1440.

To find the consumers' surplus and producers' surplus, we first need to find the equilibrium price and quantity, where the demand and supply curves intersect.

Setting the two equations equal to each other:

71 - (1/10)x = 35 + (1/20)x

Multiplying both sides by 20 to eliminate the denominators:

1420 - 2x = 700 + x

Bringing the x terms to one side:

3x = 720

Dividing both sides by 3:

x = 240

Substituting this value back into either the demand or supply equation to find the equilibrium price:

p = 71 - (1/10)(240)
p = 71 - 24
p = 47

Now that we have the equilibrium price and quantity, we can find the consumers' surplus and producers' surplus.

Consumers' surplus is the area between the demand curve and the equilibrium price up to the quantity bought. To find this, we need to calculate the area of the triangle formed by the demand curve and the price axis up to the equilibrium quantity.

Consumers' surplus = (1/2) * (240) * (71 - 47)
Consumers' surplus = (1/2) * (240) * (24)
Consumers' surplus = 2880

Producers' surplus is the area between the supply curve and the equilibrium price down to the quantity sold. To find this, we need to calculate the area of the triangle formed by the supply curve and the price axis down to the equilibrium quantity.

Producers' surplus = (1/2) * (240) * (47 - 35)
Producers' surplus = (1/2) * (240) * (12)
Producers' surplus = 1440

Therefore, the consumers' surplus is 2880 and the producers' surplus is 1440.

the equilibrium price is where supply equals demand.

71 - 1/10 x = 35 + 1/20 x
x = 240

Pmax (the y-intercept of D) is 710

Consumer surplus is thus

∫[240,710] D(x) dx

and just take it from there. Similarly for producer surplus.