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To find the producer's surplus, we first need to determine the equilibrium quantity \( x \) by equating the demand function \( D(x) \) to the supply function \( S(x) \).

### Step 1: Set Demand Equal to Supply

We are given the demand function \( D(x) = 50 - x^2 \) and the supply function \( S(x) = x^2 + 8 \). At equilibrium, demand equals supply:

\[
50 - x^2 = x^2 + 8
\]

### Step 2: Solve for \( x \)

Rearranging the equation:

\[
50 - 8 = x^2 + x^2
\]

\[
42 = 2x^2
\]

\[
x^2 = 21
\]

\[
x = \sqrt{21}
\]

### Step 3: Find the Equilibrium Price \( p \)

Substitute \( x = \sqrt{21} \) into either the demand or supply function to find the equilibrium price \( p \). Using the supply function:

\[
p = S(\sqrt{21}) = (\sqrt{21})^2 + 8 = 21 + 8 = 29
\]

### Step 4: Producer's Surplus Formula

The producer's surplus is the area between the supply curve and the equilibrium price, from \( x = 0 \) to \( x = \sqrt{21} \). Mathematically, it is given by:

\[
\text{Producer's Surplus} = \int_0^{\sqrt{21}} (p - S(x)) \, dx
\]

Substitute \( p = 29 \) and \( S(x) = x^2 + 8 \):

\[
\text{Producer's Surplus} = \int_0^{\sqrt{21}} (29 - (x^2 + 8)) \, dx = \int_0^{\sqrt{21}} (21 - x^2) \, dx
\]

### Step 5: Compute the Integral

Now, calculate the integral:

\[
\int_0^{\sqrt{21}} (21 - x^2) \, dx = \left[ 21x - \frac{x^3}{3} \right]_0^{\sqrt{21}}
\]

Substitute the limits of integration:

\[
= \left( 21\sqrt{21} - \frac{(\sqrt{21})^3}{3} \right) - (0 - 0)
\]

\[
= 21\sqrt{21} - \frac{21\sqrt{21}}{3}
\]

\[
= 21\sqrt{21} \left( 1 - \frac{1}{3} \right) = 21\sqrt{21} \times \frac{2}{3}
\]

\[
= 14\sqrt{21}
\]

Thus, the producer's surplus is \( 14\sqrt{21} \), which is approximately \( 64.2 \).

Would you like a more detailed breakdown or have any other questions? Here are some related questions:

1. What is the consumer's surplus, and how do you calculate it?
2. How would the producer's surplus change if the supply function were different?
3. What does the producer's surplus represent in an economic context?
4. Can you explain the difference between producer's and consumer's surplus?
5. How do equilibrium points affect market efficiency?

**Tip**: When solving for equilibrium, always make sure to check both the demand and supply equations at that point to confirm consistency.

If the demand function is D(x) = 50 - x^2 and the supply function is S(x) = x^2 + 8, what is the producer's surplus?

Learn how to calculate the producer's surplus given the demand function D(x) = 50 - x^2 and the supply function S(x) = x^2 + 8. This problem demonstrates the use of equilibrium points and integration to find surplus in a market setting. Suitable for college-level calculus students.

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