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To solve part (b), we need to maximize the revenue $ R(x) $, which is the product of the price $ p(x) $ and the number of spectators $ x $. The revenue function is given by:

\[
R(x) = p(x) \cdot x
\]

From part (a), we know that the demand function $ p(x) $ is:

\[
p(x) = -\frac{3}{4000}x + 28.75
\]

So, substituting $ p(x) $ into $ R(x) $:

\[
R(x) = \left(-\frac{3}{4000}x + 28.75\right) \cdot x
\]

Expanding this, we get:

\[
R(x) = -\frac{3}{4000}x^2 + 28.75x
\]

To maximize the revenue, we need to find the vertex of this quadratic function since it opens downwards (the coefficient of $ x^2 $ is negative). The $ x $-coordinate of the vertex for a quadratic function $ ax^2 + bx + c $ is given by:

\[
x = -\frac{b}{2a}
\]

Here:
- $ a = -\frac{3}{4000} $
- $ b = 28.75 $

So,

\[
x = -\frac{28.75}{2 \cdot \left(-\frac{3}{4000}\right)}
\]

Calculating this:

\[
x = \frac{28.75 \times 4000}{6} = \frac{115000}{6} \approx 19166.67
\]

Now, substitute $ x = 19166.67 $ back into $ p(x) $ to find the ticket price that maximizes revenue:

\[
p(19166.67) = -\frac{3}{4000} \cdot 19166.67 + 28.75
\]

Calculating:

\[
p(19166.67) \approx 14.375
\]

So, the ticket price should be set at approximately **$14.38** to maximize revenue.

Would you like a more detailed breakdown of any step, or have additional questions?

### Related Questions
1. What would the revenue be at the optimal ticket price?
2. How would the attendance change if the ticket price were set to maximize revenue?
3. What is the elasticity of demand at the optimal ticket price?
4. How sensitive is the revenue to small changes in ticket price near the optimal level?
5. How would increasing the stadium capacity affect the optimal ticket price?

#### Tip:
For any function maximization, finding the derivative and setting it to zero can identify critical points, which is especially useful in revenue and cost functions.

A baseball team plays in a stadium that holds 58000 spectators. With the ticket price at $10, the average attendance has been 25000. When the price dropped to $7, the average attendance rose to 29000. 

(a) Find the demand function p(x), where x is the number of spectators. 
(b) How should the ticket price be set to maximize revenue?

Learn how to maximize revenue for baseball game ticket sales by finding the demand function and setting optimal ticket prices. This problem involves calculating linear demand functions and optimizing revenue using quadratic equations. Suitable for high school students studying linear functions and revenue maximization.

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