The hourly operating cost of a plane that seats up to 381 passengers is estimated to be $11,550. If an airline charges each passenger a fare of $210 per hour of flight, find the hourly profit P it earns operating the plane as a function of the number of passengers x. [HINT: The cost function is constant (variable cost = 0).]
P(x) = 
  
Correct: Your answer is correct.
 Good work!
Specify the domain. (Enter your answer using interval notation.)

To find the hourly profit function $ P(x) $ for the airline, we start by identifying the components of the profit function.

### Step 1: Define the Revenue and Cost Functions

1. **Revenue Function (R):**  
   The revenue earned per hour is the fare charged per passenger multiplied by the number of passengers.  
   \[
   R(x) = 210x
   \]  
   where $ x $ is the number of passengers.

2. **Cost Function (C):**  
   The cost function is constant at $11,550 per hour, regardless of the number of passengers.  
   \[
   C(x) = 11,550
   \]

### Step 2: Formulate the Profit Function

The profit function $ P(x) $ is given by the difference between the revenue and the cost:  
\[
P(x) = R(x) - C(x)
\]

Substitute the expressions for $ R(x) $ and $ C(x) $:  
\[
P(x) = 210x - 11,550
\]

### Step 3: Specify the Domain of the Function

The number of passengers $ x $ can range from 0 to the maximum seating capacity of the plane, which is 381 passengers. Since $ x $ must be a non-negative integer, the domain in interval notation is:

\[
\text{Domain: } [0, 381]
\]

Thus, the hourly profit function is:

\[
\boxed{P(x) = 210x - 11,550}
\]

and the domain of $ P(x) $ is:

\[
\boxed{[0, 381]}
\]

Would you like more details on any part of the solution, or do you have any other questions?

### Related Questions:

1. What is the break-even point for the airline in terms of the number of passengers?
2. How much profit does the airline make if the plane is fully occupied with 381 passengers?
3. How would the profit function change if the fare per passenger increased to $250 per hour?
4. What is the minimum number of passengers required for the airline to earn a profit?
5. How does the profit function change if there is an additional variable cost per passenger?

**Tip:** Always remember to check the reasonableness of the domain when working with real-world functions.

Learn how to calculate the hourly profit function for an airline operating a plane that seats up to 381 passengers. Includes detailed steps on defining revenue and cost functions, and specifying the domain. Suitable for high school students studying profit functions in economics or algebra.

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