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To solve for the equilibrium price and quantity, and then calculate the consumer surplus and producer surplus, we need to follow these steps:

Given:
- Demand function: \( D(q) = 131 - \frac{1}{3}q^2 \)
- Supply function: \( S(q) = 50 + \frac{2}{3}q^2 \)

### Step 1: Find the Equilibrium Quantity and Price

At equilibrium, the quantity demanded equals the quantity supplied:
\[ D(q) = S(q) \]

So we set the two equations equal to each other:
\[ 131 - \frac{1}{3}q^2 = 50 + \frac{2}{3}q^2 \]

Combine like terms:
\[ 131 - 50 = \frac{2}{3}q^2 + \frac{1}{3}q^2 \]
\[ 81 = \frac{3}{3}q^2 \]
\[ 81 = q^2 \]
\[ q = \sqrt{81} \]
\[ q = 9 \]

So the equilibrium quantity is \( q = 9 \).

Next, find the equilibrium price by plugging \( q = 9 \) back into either the demand or supply function. Let's use the demand function:
\[ P = 131 - \frac{1}{3}(9)^2 \]
\[ P = 131 - \frac{1}{3}(81) \]
\[ P = 131 - 27 \]
\[ P = 104 \]

So the equilibrium price is \( P = 104 \) dollars.

### Step 2: Calculate the Consumer Surplus

Consumer surplus is the area of the triangle between the demand curve and the equilibrium price, up to the equilibrium quantity.

The formula for consumer surplus (CS) is:
\[ CS = \frac{1}{2} \times \text{base} \times \text{height} \]

The height is the difference between the maximum price consumers are willing to pay and the equilibrium price:
- The maximum price (when \( q = 0 \)): \( P_{max} = 131 \)
- The equilibrium price: \( P_{eq} = 104 \)

The base is the equilibrium quantity \( q = 9 \).

\[ CS = \frac{1}{2} \times 9 \times (131 - 104) \]
\[ CS = \frac{1}{2} \times 9 \times 27 \]
\[ CS = \frac{1}{2} \times 243 \]
\[ CS = 121.5 \]

### Step 3: Calculate the Producer Surplus

Producer surplus is the area of the triangle between the supply curve and the equilibrium price, up to the equilibrium quantity.

The formula for producer surplus (PS) is:
\[ PS = \frac{1}{2} \times \text{base} \times \text{height} \]

The height is the difference between the equilibrium price and the minimum price producers are willing to accept (when \( q = 0 \)):
- The minimum price (when \( q = 0 \)): \( P_{min} = 50 \)
- The equilibrium price: \( P_{eq} = 104 \)

The base is the equilibrium quantity \( q = 9 \).

\[ PS = \frac{1}{2} \times 9 \times (104 - 50) \]
\[ PS = \frac{1}{2} \times 9 \times 54 \]
\[ PS = \frac{1}{2} \times 486 \]
\[ PS = 243 \]

### Summary

- Equilibrium Quantity: \( q = 9 \)
- Equilibrium Price: \( P = 104 \)
- Consumer Surplus: \( CS = 121.5 \)
- Producer Surplus: \( PS = 243 \)

Would you like more details on any part of this solution or have any questions? Here are 5 questions you might ask next:
1. How do we derive the consumer surplus and producer surplus formulas?
2. Can you explain the concept of equilibrium in more detail?
3. What happens to consumer and producer surplus if there is a shift in the demand or supply curve?
4. How do taxes or subsidies affect the equilibrium price and quantity?
5. Can you provide more examples of calculating equilibrium and surplus in different market structures?

**Tip:** When solving for equilibrium and surplus, always ensure you correctly set the demand equal to supply, and carefully calculate the areas representing surplus to understand the welfare implications in the market.

Learn how to solve for equilibrium price and quantity in a market using supply and demand functions. This problem includes calculating consumer surplus and producer surplus, essential concepts in microeconomics. Suitable for advanced high school and college students.

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