The hourly operating cost of a certain plane, which seats up to 150 passengers, is estimated to be $8,395. If an airline charges each passenger a fare of $230 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

### 1. Profit Function $ P(x) $:
The profit function $ P(x) $ represents the total profit based on the number of passengers $ x $. To determine this, we use the following formula:

\[
P(x) = \text{Revenue} - \text{Cost}
\]

- **Revenue**: The airline charges each passenger $230 per hour. If there are $ x $ passengers, the revenue is:

\[
\text{Revenue} = 230x
\]

- **Cost**: The hourly operating cost of the plane is constant at $8,395, regardless of the number of passengers.

So, the profit function becomes:

\[
P(x) = 230x - 8395
\]

### 2. Domain of $ P(x) $:
The number of passengers $ x $ must be between 0 and 150, inclusive, because the plane seats up to 150 passengers. Therefore, the domain is:

\[
x \in [0, 150]
\]

In interval notation, this is:

\[
\text{Domain: } [0, 150]
\]

### 3. Least Number of Passengers to Make a Profit:
To find the minimum number of passengers needed to break even or start making a profit, we set $ P(x) \geq 0 $:

\[
230x - 8395 \geq 0
\]

Solving for $ x $:

\[
230x \geq 8395
\]
\[
x \geq \frac{8395}{230} \approx 36.5
\]

Since $ x $ represents the number of passengers, we round up to the nearest whole number. Therefore, the airline must carry at least **37 passengers** to make a profit.

### Summary:

1. Profit function: $ P(x) = 230x - 8395 $
2. Domain: $ [0, 150] $
3. Minimum passengers to make a profit: **37 passengers**

Do you want more details on this, or have any other questions?

---

Here are 5 related questions:
1. What happens to the profit if the number of passengers reaches 150?
2. How would the profit function change if the fare was increased to $250?
3. How would the profit function change if the operating cost increased to $10,000?
4. What would the breakeven point be if the fare was reduced to $200?
5. How much profit is made if the plane is full (150 passengers)?

**Tip:** Always round up when calculating the minimum number of items needed to cover a cost!

The hourly operating cost of a certain plane, which seats up to 150 passengers, is estimated to be $8,395. If an airline charges each passenger a fare of $230 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?

Calculate the hourly profit for an airline with a plane that seats up to 150 passengers, charging $230 per hour per passenger and operating at a constant cost of $8,395. Learn how to create a linear profit function, determine the domain, and find the minimum number of passengers required to make a profit. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation