To find the profit $P(x)$ as a function of $x$, the number of tickets sold, we'll first determine the price per ticket as a function of $x$, then calculate the revenue and costs, and finally determine the profit function.

Step 1: Determine the price function $p(x)$

Given the dataTo find the profit $P(x)$ as a function of $x$, the number of tickets sold, we'll first determine the price per ticket as a function of $x$, then calculate the revenue and costs, and finally determine the profit function.

Step 1: Determine the price function $p(x)$

Given the

• When $p = 4$, $x = 45,000$
• When $p = 5$, $x = 40,000$

Assuming a linear demand function $p(x) = mx + b$, we can find the slope $m$ using the two points:

$m = \frac{5 - 4}{40,000 - 45,000} = \frac{1}{-5,000} = -\frac{1}{5,000}$

Using the point-slope form of the line equation:

$p(x) - 4 = -\frac{1}{5,000}(x - 45,000)$

Simplifying, we find:

$p(x) = -\frac{1}{5,000}x + 13$

Step 2: Calculate Revenue and Costs

Revenue $R(x)$ is the price per ticket multiplied by the number of tickets sold:

$R(x) = p(x) \times x = \left(-\frac{1}{5,000}x + 13\right) \times x = -\frac{1}{5,000}x^2 + 13x$

Total Costs $C(x)$ consist of variable costs and fixed costs:

$C(x) = 0.10x + 95,000$

Step 3: Determine the Profit Function $P(x)$

Profit is given by the difference between revenue and total costs:

$P(x) = R(x) - C(x) = \left(-\frac{1}{5,000}x^2 + 13x\right) - (0.10x + 95,000)$

Simplifying the profit function:

$P(x) = -\frac{1}{5,000}x^2 + 13x - 0.10x - 95,000$

$P(x) = -\frac{1}{5,000}x^2 + 12.90x - 95,000$

Final Answer:

$\boxed{P(x) = -\frac{1}{5,000}x^2 + 12.90x - 95,000}$

Would you like more details or have any questions?

Related Questions:

1. What is the maximum profit, and how many tickets need to be sold to achieve it?
2. How does the break-even point change with different fixed costs?
3. What would the profit function look like if the variable costs increased to $0.20 per ticket?
4. How would the demand function change if the ticket sales were 50,000 at $6 per ticket?
5. What is the impact on profit if a discount of $1 per ticket is applied for every 10,000 tickets sold?

Tip: When dealing with profit functions, always check for the vertex of the parabola to find the maximum profit point.