1.) Find the producers' surplus if the supply function is: S(q) = q^7/2+3q^5/2 + 54. Assume the supply and demand are in equilibrium at q= 25.

2.) S(q) = q^2 + 12q and D(q) = 900 - 18q - q^2

The point at which the supply and demand are equilibrium is (15, 405).

a.) Find the consumers' surplus
b.) Find the producers' surplus

To find the producers' surplus, we first need to understand what it represents. Producers' surplus is a measure of the economic benefit that producers receive when they sell their goods or services at a price higher than their cost of production.

The formula for producers' surplus is given by the integral of the supply curve from the equilibrium quantity to the quantity being considered.

1.) For the first question, we are given the supply function S(q) = q^(7/2) + 3q^(5/2) + 54 and the equilibrium quantity q = 25.

To find the producers' surplus, we need to calculate the area between the supply curve and the equilibrium quantity.

Step 1: Calculate the supply curve at the equilibrium quantity.
At q = 25, the supply function becomes S(25) = 25^(7/2) + 3(25)^(5/2) + 54.

Step 2: Calculate the integral of the supply function between the equilibrium quantity and the quantity being considered.
We need to integrate the supply function S(q) from q = 25 to q = 25.

The producers' surplus is given by the integral of S(q) between these two quantities.

2.) For the second question, we are given the supply function S(q) = q^2 + 12q and the demand function D(q) = 900 - 18q - q^2, as well as the equilibrium point (15, 405).

a.) To find the consumers' surplus, we need to calculate the area between the demand curve and the equilibrium quantity.

Step 1: Calculate the demand curve at the equilibrium quantity.
At q = 15, the demand function becomes D(15) = 900 - 18(15) - (15)^2.

Step 2: Calculate the integral of the demand function between the equilibrium quantity and the quantity being considered.
We need to integrate the demand function D(q) from q = 15 to q = 15.

The consumers' surplus is given by the integral of D(q) between these two quantities.

b.) To find the producers' surplus, we follow the same steps as in the first question. The producers' surplus is given by the integral of S(q) between the equilibrium quantity and the quantity being considered. In this case, the quantity being considered is also q = 15.

By calculating the integral of S(q) between q = 15 and q = 15, we can find the producers' surplus.

Please note that for both questions, the integral calculation may involve some algebraic manipulations and integration techniques.

1.) To find the producers' surplus, we need to calculate the area above the supply curve and below the equilibrium price (q=25).

The integral formula for producers' surplus is:
∫[E, q] S(q) dq
where E is the equilibrium quantity and q is the quantity of the supply curve.

Given the supply function: S(q) = q^7/2 + 3q^5/2 + 54, we can integrate it to find the producers' surplus.

Producers' surplus = ∫[E, q] (q^7/2 + 3q^5/2 + 54) dq

Since the equilibrium quantity E is not given, we'll assume it is q=25 as stated in the question. We can then calculate the producers' surplus.

Producers' surplus = ∫[25, q] (q^7/2 + 3q^5/2 + 54) dq

To evaluate this integral, we can solve it term by term:
∫[25, q] q^7/2 dq + ∫[25, q] 3q^5/2 dq + ∫[25, q] 54 dq

Integrating each term:
(2/9) q^(9/2) |[25, q] + (2/7) q^(7/2) |[25, q] + 54q |[25, q]

Evaluating each term:
[(2/9) q^(9/2) - (2/9) (25)^(9/2)] + [(2/7) q^(7/2) - (2/7) (25)^(7/2)] + (54q - 54(25))

Simplifying:
[(2/9) q^(9/2) - 4248/9] + [(2/7) q^(7/2) - 6250/7] + 54q - 1350

Replacing q with 25:
(2/9) (25)^(9/2) - 4248/9 + (2/7) (25)^(7/2) - 6250/7 + 54(25) - 1350

Calculating this expression will give us the producers' surplus.