The hourly operating cost of a certain plane, which seats up to 150 passengers, is estimated to be $7,810. If an airline charges each passenger a fare of $220 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*) =

Specify the domain. (Enter your answer using interval notation.)

What is the least number of passengers it must carry to make a profit? (Round your answer up to the nearest passenger.)

passengers'

The hourly profit $ P(x) $ for the airline can be modeled as the difference between the total revenue from the passengers and the total cost of operating the plane.

### Step 1: Define the revenue function.
The airline charges each passenger a fare of \$220 per hour. Therefore, the total revenue $ R(x) $, where $ x $ is the number of passengers, is given by:
\[
R(x) = 220x
\]

### Step 2: Define the cost function.
The hourly operating cost of the plane is given as \$7,810, which is constant regardless of the number of passengers. Therefore, the total cost $ C(x) $ is:
\[
C(x) = 7,810
\]

### Step 3: Define the profit function.
The profit is the difference between the total revenue and the total cost. Thus, the profit function $ P(x) $ is:
\[
P(x) = R(x) - C(x) = 220x - 7,810
\]

### Step 4: Specify the domain.
The number of passengers $ x $ can range from 0 to 150, since the plane has a maximum seating capacity of 150 passengers. Therefore, the domain of the function is:
\[
[0, 150]
\]

### Step 5: Find the least number of passengers to make a profit.
To make a profit, $ P(x) $ must be greater than 0:
\[
220x - 7,810 > 0
\]
Solving for $ x $:
\[
220x > 7,810
\]
\[
x > \frac{7,810}{220} \approx 35.5
\]
Since the number of passengers must be a whole number, the airline must carry at least 36 passengers to make a profit.

### Summary:
- Profit function: $ P(x) = 220x - 7,810 $
- Domain: $ [0, 150] $
- Least number of passengers to make a profit: 36 passengers

Would you like more details or have any questions?

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**Related Questions:**
1. What would the profit be if the airline carries the maximum 150 passengers?
2. How would the profit change if the fare per passenger increased to \$250 per hour?
3. What is the break-even point for the number of passengers at which the airline neither profits nor loses money?
4. How does the profit function change if the operating cost increases by 10%?
5. If the plane operates for 5 hours, how would the total profit function look like?

**Tip:** Understanding how to break down a problem into revenue and cost functions can help in various business and economic applications.

The hourly operating cost of a certain plane, which seats up to 150 passengers, is estimated to be $7,810. If an airline charges each passenger a fare of $220 per hour of flight, find the hourly profit P 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 find the profit function for an airline operating a plane that seats 150 passengers. The problem involves calculating the profit as a function of the number of passengers and determining the least number of passengers needed to make a profit. This solution includes key algebra concepts and profit calculation. Suitable for Grades 9-12.

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