Problem 41 - Precalculus with Limits

Question

Airplane Speed An airplane flying into a headwind travels the 1800 -mile flying distance between Pittsburgh, Pennsylvania, and Phoenix, Arizona, in 3 hours and 36 minutes. On the return flight, the airplane travels this distance in 3 hours. Find the airspeed of the plane and the speed of the wind, assuming that both remain constant.

Step-by-Step Solution

Answer

The airspeed of the plane is 550 mph and the wind speed is 50 mph.

Step 1: Identify the equations

There will be two different speeds for the plane: It's speed while travelling to Phoenix and it's speed while returning. The speed is the distance covered / time taken. Let's denote the airspeed of the plane as \( p \) and the speed of the wind as \( w \). The plane's ground speed when flying into the headwind (to Phoenix) is given by \( p - w \), while the ground speed when flying with the wind (coming back) can be expressed as \( p + w \). Hence you can setup two equations: \n1) \( p - w = \frac{Distance\_to\_Phoenix}{Time\_to\_Phoenix} \).\n2) \( p + w = \frac{Distance\_from\_Phoenix}{Time\_from\_Phoenix} \).

Step 2: Substitute the values into equations

Substitute the values of the distances and times into these equations. Convert 3 hours and 36 minutes to an equivalent value in hours, which is 3.6. You will have the following equations: \n1) \( p - w = \frac{1800}{3.6} \), \n2) \( p + w = \frac{1800}{3} \). These can be simplified to \n1) \( p - w = 500 \),\n2) \( p + w = 600 \).

Step 3: Solve the equations

Sum up the two equations \(500 + 600 = p - w + p + w\). This gives \(1100 = 2p + 0\). Simplifying this, you get \( p = 550 \) mph. \nThen, plug \( p = 550 \) into the first equation to get the value of \( w \). This gives:\n \( 550 - w = 500 \) \nThen solving for \( w \) gives:\n \( w = 550 - 500 = 50 mph\).

Key Concepts

Airspeed and Wind Speed

Understanding the impact of wind on an airplane's journey is fundamental to solving flight speed problems. Airspeed refers to the speed of an aircraft relative to the air around it. However, aircraft also encounter wind, which can either slow them down (headwind) or speed them up (tailwind).

The speed of the airplane, unaffected by the wind, is known as the airspeed (\( p \) in problems), while the wind speed is (\( w \) in problems). In calculations, when a plane faces a headwind, the effective ground speed is 'airspeed minus wind speed' (\( p - w \)); conversely, if the plane is assisted by a tailwind, the effective ground speed is 'airspeed plus wind speed' (\( p + w \)). Understanding this relationship is crucial in determining the actual flight time for a given distance.

To solve a problem involving airspeed and wind speed, it’s essential to isolate these variables and create equations that correspond to the different speeds experienced in opposite directions.

Distance Time Speed Relationship

The relationship between distance, time, and speed is a key concept in motion problems and can be expressed by the formula: speed = distance/time. This relationship allows for the determination of any one of these variables if the other two are known.

When an airplane travels a fixed distance, the time taken for the journey can vary based on its speed. The duration of a flight is inversely proportional to the speed: the faster the speed, the shorter the time needed for the same distance, and vice versa.

Applying the Formula

When given the distance and the time, like in the airplane speed problem, we can determine the ground speed for each leg of the journey. In the context of the given example, the distance remains constant at 1800 miles, but the time changes due to the influence of wind, which ultimately affects the ground speed.

Solving System of Equations

A system of equations consists of two or more equations with the same set of unknowns. Solving such a system means finding the values of the variables that make all the equations true simultaneously.

In the context of the given airplane speed problem, we encounter a system of two equations, where the unknowns are the airspeed and wind speed. To solve these equations, one can use various methods like substitution, elimination, or matrix techniques. The most efficient method here is to use the elimination method, adding or subtracting equations to eliminate one of the variables.

Working with the Equations

By summing the two equations that represent the respective speeds with and against the wind, we can eliminate the wind speed and solve for the airspeed. Then, we can substitute this value into one of the original equations to find the wind speed.

Converting Time Units

To work with the distance time speed relationship accurately, it's necessary to have all units consistent. Time conversions are often required in flight speed problems, as time can be presented in hours and minutes, while calculations are simplified when the time is solely in hours.

Conversion is straightforward: 60 minutes equal one hour. Thus, to convert minutes into a fraction of an hour, divide the minutes by 60. For example, in our airplane speed problem, 36 minutes is converted to hours by dividing 36 by 60, resulting in 0.6 hours. A clear understanding of this conversion process is essential for accurate calculations and solving time-related problems effectively.

Practical Application

When solving the airplane problem, one must combine the 3 hours with the the minutes converted to hours (0.6), which sums up to 3.6 hours total flying time. This accurate conversion ensures that the equations used to solve for airspeed and wind speed are correct.