Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

An airplane flying into a headwind travels the 1800 -mile flying distance between New York City and Albuquerque, New Mexico, in 3 hours and 36 minutes. On the return flight, the same distance is traveled 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 approximately 545.45 mph and the wind speed, approximately 45.455 mph.

Step 1: Setup the Equations

To start with, let's denote two variables: p for the airspeed of the plane and w for the speed of the wind. Since flying into a headwind (from New York to Albuquerque) is slower, we have the equation: \(1800 = (p-w) * 3.6\) . Similarly, flying with the wind (from Albuquerque back to New York) is faster, so we have the equation \(1800 = (p + w) * 3\). These are two equations with two variables (p and w).

Step 2: Solving the system of equations

Adding the two equations gives: \(1800 + 1800 = (p + w) * 3 + (p - w) * 3.6\), which simplifies to give \(3600 = 3p + 3.6p\), so, solving for p, \(p = 3600 / 6.6 ≈ 545.45\). Now, you can use either of the original equations to solve for w. Let's use the second one. Plugging p value into it, we get \(1800 = (545.45 + w) * 3\), solving for w gives \(w ≈ 45.455\)

Step 3: Check the Solution

It's always a good idea to check your solution. Substitute p and w back to the two original equations and check if they are technically equal.

Key Concepts

Problem Solving

When faced with a mathematical problem, the first step is to clearly understand what is being asked.
In this exercise, we need to find the airspeed of a plane and the wind speed.
This involves creating equations based on information provided. Key elements to consider include:

• Identifying the unknowns: Here, we need to identify two variables - the airspeed of the plane and the speed of the wind.
• Setting up equations: Based on the problem, we create equations that describe the relationship between these variables.

For this problem, we use the concept of relative speed, which involves both the plane speed and the wind speed impacting the overall travel time.
By interpreting the journey into and against the wind, we can form the necessary equations to solve the problem.
Once the system of equations is established, we can proceed to solve for the unknowns.

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns.
In our case, we have two equations for two unknowns: the airspeed of the plane and the speed of the wind.The equations are:

• Into the wind: \(1800 = (p-w) \cdot 3.6\)
• With the wind: \(1800 = (p+w) \cdot 3\)

These equations represent the relationship between the distance, speed, and time for the flights to and from Albuquerque.
By using algebraic techniques such as adding equations or substitution, we can find the values of \(p\) and \(w\).
Combining these two equations allows us to eliminate one variable and solve for the other.
This step is crucial for arriving at an accurate solution in a systematic way.

Speed Calculation

Calculating speed involves understanding the relationship between distance, time, and speed.
The formula is straightforward: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]In this problem, the distance (1800 miles) remains constant, while the time varies based on wind direction.

• When flying into the wind, the plane travels slower, indicating a decreased net speed caused by the opposite force of the wind.
• Conversely, flying with the wind increases the speed due to the additional push from the wind.

By substituting the known distance and time into the formula, you can determine the effective speed (\(p-w\) or \(p+w\)).
Once speeds relative to the wind are known, they can be used to find the actual airspeed of the plane and the speed of the wind itself.
This approach is practical for understanding how conditions like wind can impact travel time and speed.