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Let's break down the problem based on the given information:

1. **Operating cost**: The plane's hourly operating cost is $8,165.
2. **Fare per passenger**: The airline charges each passenger $230 per hour.
3. **Passengers (variable)**: Let $ x $ represent the number of passengers, where the maximum number of passengers is 150.
4. **Profit function $ P(x) $**: We are asked to determine the profit function $ P(x) $ as a function of the number of passengers.

### Step 1: Deriving the profit function $ P(x) $

- The revenue the airline earns from $ x $ passengers is:
  \[
  \text{Revenue} = 230x
  \]
  
- The cost of operating the plane is a constant $ 8,165 $ (since variable costs are assumed to be 0).

- Therefore, the profit function $ P(x) $ is:
  \[
  P(x) = 230x - 8165
  \]

### Step 2: Specifying the domain

The number of passengers $ x $ must be between 0 and 150, since the plane seats up to 150 passengers. Therefore, the domain is:
\[
[0, 150]
\]
(in interval notation).

### Step 3: Finding the least number of passengers required to make a profit

- To find the least number of passengers required to make a profit, set $ P(x) \geq 0 $, meaning the profit must be at least 0:
  \[
  230x - 8165 \geq 0
  \]
  
  Solving for $ x $:
  \[
  230x \geq 8165
  \]
  \[
  x \geq \frac{8165}{230} \approx 35.5
  \]
  
- Since the number of passengers must be a whole number, round up to the nearest passenger:
  \[
  x \geq 36
  \]

Therefore, the least number of passengers required to make a profit is **36** passengers.

---

### Summary:
- Profit function: $ P(x) = 230x - 8165 $
- Domain: $ [0, 150] $
- Least number of passengers to make a profit: **36** passengers.

Let me know if you'd like more details or have any questions.

Here are 5 related questions you might find interesting:
1. How would the profit function change if the operating cost were reduced by 10%?
2. What happens if the fare per passenger increases by $50?
3. How would the break-even point change if the plane could seat 200 passengers?
4. What is the maximum possible profit the airline can earn if the plane is fully booked?
5. How would the profit function change if there was a variable cost per passenger?

**Tip:** Always analyze profit and revenue equations to see where break-even points lie—this is key to maximizing efficiency.

The hourly operating cost of a certain plane, which seats up to 150 passengers, is estimated to be $8,165. If an airline charges each passenger a fare of $230 per hour of flight, find the hourly profit P(x) it earns operating the plane as a function of the number of passengers x. Specify the domain. What is the least number of passengers it must carry to make a profit?

Learn how to calculate the profit function for an airline with a plane that seats up to 150 passengers and has an operating cost of $8,165. This problem involves determining the least number of passengers required to make a profit, using a fare of $230 per passenger. Suitable for high school math students in Grades 9-12.

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