The demand function for a certain brand of compact disc is given by
p(x) = - 0.01x2 - 0.3x + 11
The supply function for the compact discs of is given by
p(x) = 0.01x2 + 0.2x + 4
where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the producers' surplus if the wholesale market price is set at the equilibrium price. 

To determine the producers' surplus when the wholesale market price is set at the equilibrium price, we first need to find the equilibrium price and quantity. 

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

The equilibrium price and quantity occur where the demand function equals the supply function. Thus, we set the two functions equal to each other and solve for $ x $:

\[ -0.01x^2 - 0.3x + 11 = 0.01x^2 + 0.2x + 4 \]

First, combine like terms:

\[ -0.01x^2 - 0.3x + 11 = 0.01x^2 + 0.2x + 4 \]

\[ -0.01x^2 - 0.01x^2 - 0.3x - 0.2x + 11 - 4 = 0 \]

\[ -0.02x^2 - 0.5x + 7 = 0 \]

Multiply through by $-1$ to simplify:

\[ 0.02x^2 + 0.5x - 7 = 0 \]

Next, solve this quadratic equation using the quadratic formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, where $ a = 0.02 $, $ b = 0.5 $, and $ c = -7 $:

\[ x = \frac{-0.5 \pm \sqrt{(0.5)^2 - 4(0.02)(-7)}}{2(0.02)} \]

\[ x = \frac{-0.5 \pm \sqrt{0.25 + 0.56}}{0.04} \]

\[ x = \frac{-0.5 \pm \sqrt{0.81}}{0.04} \]

\[ x = \frac{-0.5 \pm 0.9}{0.04} \]

This gives us two solutions:

\[ x = \frac{0.4}{0.04} = 10 \]

\[ x = \frac{-1.4}{0.04} = -35 \]

Since a negative quantity doesn't make sense in this context, we discard $ x = -35 $. So, the equilibrium quantity $ x $ is 10 thousand units.

To find the equilibrium price $ p $, substitute $ x = 10 $ back into either the demand or supply function. We'll use the supply function $ p(x) = 0.01x^2 + 0.2x + 4 $:

\[ p(10) = 0.01(10)^2 + 0.2(10) + 4 \]

\[ p(10) = 0.01(100) + 2 + 4 \]

\[ p(10) = 1 + 2 + 4 \]

\[ p(10) = 7 \]

So, the equilibrium price is $7 per unit.

### Step 2: Determining the Producers' Surplus

Producers' surplus is the area above the supply curve and below the equilibrium price. It can be calculated as follows:

\[ \text{Producers' Surplus} = \int_{0}^{10} [p_e - S(x)] \, dx \]

where $ p_e $ is the equilibrium price, and $ S(x) $ is the supply function.

Here, $ p_e = 7 $ and $ S(x) = 0.01x^2 + 0.2x + 4 $:

\[ \text{Producers' Surplus} = \int_{0}^{10} [7 - (0.01x^2 + 0.2x + 4)] \, dx \]

\[ = \int_{0}^{10} (7 - 0.01x^2 - 0.2x - 4) \, dx \]

\[ = \int_{0}^{10} (3 - 0.01x^2 - 0.2x) \, dx \]

Now, integrate each term separately:

\[ \int_{0}^{10} 3 \, dx = 3x \bigg|_{0}^{10} = 3(10) - 3(0) = 30 \]

\[ \int_{0}^{10} 0.01x^2 \, dx = 0.01 \int_{0}^{10} x^2 \, dx = 0.01 \left( \frac{x^3}{3} \bigg|_{0}^{10} \right) = 0.01 \left( \frac{10^3}{3} - 0 \right) = 0.01 \left( \frac{1000}{3} \right) = \frac{10}{3} \approx 3.33 \]

\[ \int_{0}^{10} 0.2x \, dx = 0.2 \int_{0}^{10} x \, dx = 0.2 \left( \frac{x^2}{2} \bigg|_{0}^{10} \right) = 0.2 \left( \frac{10^2}{2} - 0 \right) = 0.2 \left( \frac{100}{2} \right) = 0.2 \times 50 = 10 \]

Combining these results:

\[ \text{Producers' Surplus} = 30 - 3.33 - 10 = 16.67 \]

Thus, the producers' surplus is approximately $16,670 (since the units are in thousands).

Do you have any questions or need further details on any part of the solution?

Here are 8 related questions you might find interesting:
1. How does the demand function change if the price elasticity increases?
2. What happens to the equilibrium price if the supply curve shifts to the right?
3. How would a tax on producers affect the equilibrium price and quantity?
4. What is consumer surplus, and how is it calculated in this scenario?
5. How does a subsidy to producers influence the supply curve?
6. What are the welfare implications of setting a price floor above the equilibrium price?
7. How can we determine the deadweight loss due to price controls in this market?
8. What factors might cause the demand curve to shift in the compact disc market?

**Tip:** When dealing with equilibrium problems, always ensure to carefully handle the units and interpret the physical meaning of the results.

Learn how to calculate producers' surplus at equilibrium price and quantity in economics using integration. Detailed step-by-step solution involving the equilibrium price, quantity, demand function, and supply function.

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