Problem 59 - Precalculus

Question

A direct flight on Delta Air Lines from Atlanta to Paris is 4000 miles and takes approximately 8 hours going east (Atlanta to Paris) and 10 hours going west (Paris to Atlanta). Although the plane averages the same airspeed, there is a headwind while traveling west and a tailwind while traveling east, resulting in different air speeds. What is the average air speed of the plane, and what is the average wind speed?

Step-by-Step Solution

Answer

The average airspeed of the plane is 450 mph, and the average wind speed is 50 mph.

Step 1: Define Variables

Let's assign variables to the unknowns. Let \( p \) represent the average air speed of the plane (in miles per hour) and \( w \) the average wind speed (in miles per hour).

Step 2: Express Speeds in Terms of Variables

When flying east from Atlanta to Paris, the plane has a tailwind, making its effective speed \( p + w \). On the way west, from Paris to Atlanta, there is a headwind, making the effective speed \( p - w \).

Step 3: Write Equations for Each Trip

Use the formula for distance: \[ \text{Distance} = \text{Speed} \times \text{Time} \]So for the eastbound trip, we have:\[ 4000 = (p + w) \times 8 \]And for the westbound trip:\[ 4000 = (p - w) \times 10 \]

Step 4: Solve the Equations for Airspeed and Wind Speed

For the eastbound equation: \[ 4000 = (p + w) \times 8 \] Solving for \( p + w \):\[ p + w = 500 \]For the westbound equation: \[ 4000 = (p - w) \times 10 \]Solving for \( p - w \):\[ p - w = 400 \]

Step 5: Solve the System of Equations

Add the two equations:\[ (p + w) + (p - w) = 500 + 400 \]\[ 2p = 900 \]\[ p = 450 \]Now substitute back to find \( w \):\[ p + w = 500 \]\[ 450 + w = 500 \]\[ w = 50 \]

Step 6: Review Solution

We have found that the average airspeed of the plane (\( p \)) is 450 mph and the average wind speed (\( w \)) is 50 mph. Verify these results satisfy both conditions:\- Eastbound: \( (450 + 50) \times 8 = 4000 \)\- Westbound: \( (450 - 50) \times 10 = 4000 \)

Key Concepts

Distance-Speed-Time Relationship

When dealing with distances covered by a vehicle over time, the distance-speed-time relationship plays a pivotal role. This relationship is expressed through the formula: \[ \text{Distance} = \text{Speed} \times \text{Time} \] This formula helps determine any one of these three variables, given the other two. In our problem, we use this relationship to find out the effective speed of the airplane both when moving east with a tailwind and heading west with a headwind. To solve such problems, it's crucial to recognize whether the speed given is influenced by any external factors such as wind, which will either increase or decrease the effective speed. In this exercise, we see how wind plays an essential role, which is explained further in the next sections. By setting up equations based on this relationship, understanding and resolving the impacts of external factors become straightforward.

Headwind and Tailwind Effects

Wind can drastically change the effective speed of an airplane, acting either as headwind or tailwind. These effects play significant roles in the calculation of average airspeed. - **Tailwind**: When traveling east (from Atlanta to Paris), the plane benefits from a tailwind, meaning the wind is moving in the same direction as the plane. This increases the effective speed of the plane. Therefore, the combined speed is the plane's airspeed plus the wind speed \( (p + w) \). - **Headwind**: Conversely, when the plane flies west (from Paris to Atlanta), it faces a headwind, as the wind opposes the direction of flight. This results in a decrease of the effective speed, thus calculated as the plane's airspeed minus the wind speed \( (p - w) \). These calculations highlight how external factors such as wind can require adjustments in our calculations to predict or analyze the actual travel time accurately.

Average Speed Calculations

Average speed is a fundamental calculation in determining travel efficiency and airflow dynamics. It's crucial to differentiate between airspeed and effective speed due to environmental factors, such as wind, that might alter travel circumstances. To compute the average airspeed of the plane, we use the data from both travel directions. In our exercise, the total travel time is different in each direction because of the wind affecting the effective speed. By understanding both the headwind and tailwind contributions—calculated from their respective equations \( p + w = 500 \) and \( p - w = 400 \)—we find the: - **Average Airspeed (p)**: Achieved by solving the system of equations \[ 2p = 900 \] \[ p = 450 \text{ mph} \]- **Average Wind Speed (w)**: Once airspeed is known, calculating wind speed becomes straightforward \[ w = 50 \text{ mph} \]By resolving such systems, you can accurately find average speeds, offering insights into flight planning and efficient travel.