Problem 46 - Prealgebra

Question

An airplane flew from Boston through Chicago to Denver. The distance from Boston to Chicago was 1015 miles, and the distance from Chicago to Denver was 1011 miles. The plane traveled for 3.5 hours and carried 285 passengers. Suppose it costs \(\$ 5685\) per hour to operate the airplane. Find the cost per person per hour for the flight.

Step-by-Step Solution

Answer

The cost per person per hour for the flight is approximately \$20.05.

Step 1: Calculate the Total Flying Time

The problem states that the airplane traveled for 3.5 hours. This is the total flying time for which the cost needs to be calculated.

Step 2: Calculate the Total Cost of the Flight

Multiply the total flying time by the hourly operating cost to find the total cost of the flight: \[ Total\ Cost = 3.5\ hours \times \\(5685\ /\ hour = \\)19997.5 \]

Step 3: Calculate the Cost per Passenger

Divide the total cost of the flight by the number of passengers to find the cost per passenger:\[ Cost\ per\ Passenger = \frac{\\(19997.5}{285\ passengers} = \\)70.17 \]

Step 4: Calculate the Cost per Passenger per Hour

Divide the cost per passenger by the number of hours to determine the cost per passenger per hour:\[ Cost\ per\ Passenger\ per\ Hour = \frac{\\(70.17}{3.5\ hours} = \\)20.05 \]

Key Concepts

Understanding Operating Cost

The operating cost of an airplane is the total amount of money it takes to run the plane for a specific period of time. This cost often includes fuel expenses, maintenance, crew salaries, and other operating expenses. In this exercise, it's given that the airplane costs $5685 per hour to operate. Knowing this figure is crucial, as it forms the core of any cost-related calculations for flights.

To compute the total operating cost for a flight, you multiply the hourly operating cost by the total flying time. In practical scenarios, airlines calculate these expenses meticulously, as they impact the pricing of flights and ultimately, the profitability. For example, if an airplane travels for 3.5 hours, as this exercise states, the total cost would be computed as:

• Total Operating Cost = Hourly Operating Cost × Total Hours of Flight
• Total Operating Cost = $5685 × 3.5
• Total Operating Cost = $19997.5

This total cost forms the base for further calculations, such as determining the cost per passenger.

Calculating Passenger Cost

Passenger cost refers to the share of each passenger in covering the total operating expenses of a flight. Airlines need to determine this cost to set ticket prices that are competitive while ensuring cost recovery.

To find the passenger cost, you divide the total operating cost by the number of passengers on board. This gives the average cost each passenger indirectly pays for the flight. In our exercise, with a total cost of $19997.5 and 285 passengers:

• Passenger Cost = Total Operating Cost ÷ Number of Passengers
• Passenger Cost = 19997.5 ÷ 285
• Passenger Cost = $70.17

This calculation helps you understand how much, on average, each passenger contributes to covering the flight's operating costs.

Time Calculation and Its Importance

Time calculation in air travel is crucial for determining costs effectively. The duration of the flight directly impacts operating costs since longer flights generally entail higher expenses.

In this scenario, calculating the cost per passenger per hour requires understanding the role of time. By dividing the cost each passenger pays by the hours flown, you get a clearer picture of the expense rate per passenger.

• Cost per Passenger per Hour = Passenger Cost ÷ Total Flying Hours
• Cost per Passenger per Hour = $70.17 ÷ 3.5
• Cost per Passenger per Hour = $20.05

This metric provides insights into how efficiently the flight generates revenue relative to the time enduring the operational costs. Both airlines and passengers benefit from this understanding, as it allows airlines to optimize schedules and routes, while passengers can better appreciate the value of their ticket prices.