​D(x) is the​ price, in dollars per​ unit, that consumers are willing to pay for x units of an​ item, and​ S(x) is the​ price, in dollars per​ unit, that producers are willing to accept for x units. Find ​(a​) the equilibrium​ point, ​(b​) the consumer surplus at the equilibrium​ point, and ​(c​) the producer surplus at the equilibrium point.

D(x)=(x-5)^2, S(x)=(x^2)+4x+11

To solve this problem, we need to find the equilibrium point where the quantity supplied equals the quantity demanded, and then calculate the consumer and producer surplus at that equilibrium point.

### Given:
- Demand function: \( D(x) = (x-5)^2 \)
- Supply function: \( S(x) = x^2 + 4x + 11 \)

### (a) Finding the Equilibrium Point

The equilibrium point occurs where the demand equals the supply:

\[
D(x) = S(x)
\]

Substituting the given functions:

\[
(x - 5)^2 = x^2 + 4x + 11
\]

Let's solve this equation.

### (b) Consumer Surplus at the Equilibrium Point

The consumer surplus is calculated as:

\[
\text{Consumer Surplus} = \int_{0}^{x_e} D(x) \, dx - (p_e \times x_e)
\]

Where \( x_e \) and \( p_e \) are the equilibrium quantity and price, respectively.

### (c) Producer Surplus at the Equilibrium Point

The producer surplus is calculated as:

\[
\text{Producer Surplus} = (p_e \times x_e) - \int_{0}^{x_e} S(x) \, dx
\]

Now let's solve these steps one by one. I'll start with solving for the equilibrium point.### (a) Equilibrium Point

The equilibrium quantity is \( x_e = 1 \).

To find the equilibrium price \( p_e \), substitute \( x_e = 1 \) into either the demand or supply function:

\[
p_e = D(1) = S(1)
\]

Let's calculate \( p_e \).The equilibrium price is \( p_e = 16 \). 

So, the equilibrium point is \((x_e, p_e) = (1, 16)\).

### (b) Consumer Surplus

To find the consumer surplus, we use the formula:

\[
\text{Consumer Surplus} = \int_{0}^{1} D(x) \, dx - (16 \times 1)
\]

Let's calculate this.The consumer surplus at the equilibrium point is \(\frac{13}{3}\) or approximately 4.33 dollars.

### (c) Producer Surplus

To find the producer surplus, we use the formula:

\[
\text{Producer Surplus} = (16 \times 1) - \int_{0}^{1} S(x) \, dx
\]

Let's calculate this now.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can calculate the producer surplus by using the formula:

\[
\text{Producer Surplus} = (16 \times 1) - \int_{0}^{1} (x^2 + 4x + 11) \, dx
\]

Would you like help with any further questions or calculations?

Here are some related questions to consider:

1. What is the economic interpretation of consumer and producer surplus?
2. How would the equilibrium point change if the supply function was modified?
3. What happens to the consumer and producer surplus if the demand function changes?
4. Can the equilibrium point be found if both functions are linear?
5. How would taxes impact the equilibrium point and surplus calculations?
6. What role does elasticity play in determining the surplus?
7. How can surplus values help businesses in decision-making?
8. What would be the impact of a price floor or ceiling on the surplus?

**Tip:** Understanding the area under the demand and supply curves is key to calculating surplus effectively.

Learn how to find the equilibrium point, consumer surplus, and producer surplus using integral calculus in economics. Detailed step-by-step solution for demand and supply functions D(x) = (x-5)^2 and S(x) = x^2 + 4x + 11. Suitable for advanced college-level students studying economics.

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