Question

It took 3 hours for a plane, flying against the wind, to travel 900 miles from Alabama to Minnesota. The "ground speed" of the plane is 300 miles per hour. On the return trip, the flight took only 2 hours with a ground speed of 450 miles per hour. During both flights the speed and the direction of the wind were the same. The plane's speed decreases or increases because of the wind as the verbal model below shows. Speed in still air \(]-[\text { Wind speed }]=\) Ground speed against wind Speed in still air \(+\text { Wind speed }]=\) Ground speed with wind What was the speed of the plane in still air? What was the speed of the wind?

Short answer

The speed of the plane in still air was 375 miles per hour. The speed of the wind was 75 miles per hour.

Step-by-step solution

Set Up Equations

From the problem, we know that the ground speed of the plane against the wind is 300 miles per hour and with the wind is 450 miles per hour. Let's denote the speed of the plane in still air as P (in mph) and the speed of the wind as W (in mph). We can set up two equations based on the given model. The first one is P - W = 300 (the speed of the plane against the wind) and the second one is P + W = 450 (the speed of the plane with the wind).

Solve Equations

Now we can use the method of elimination to solve these equations. If we add these two equations we get: Simplify to get 2P = 750, thus P = 375 mph.

Find Wind Speed

Substitute P = 375 mph into the first equation, we get 375 - W = 300, thus W = 75 mph.

Key concepts

Solving Systems of Equations

Understanding how to solve systems of equations is essential in algebra and has practical applications, such as determining the speed of a plane affected by wind. In this context, we deal with two equations that share variables.

For the plane speed problem, we represent the plane's speed in still air as variable P, and the wind speed as variable W. The first equation is derived from the plane's ground speed against the wind:
\( P - W = 300 \).
The second equation comes from the ground speed with the wind:
\( P + W = 450 \).

To solve these, one common method is by elimination, which involves combining the equations to eliminate one of the variables. Adding them together cancels out W, allowing us to find P. Once P is known, we substitute it back into either equation to determine W. It's like solving a puzzle: align the pieces (equations) in such a way to see the complete picture (solution).

Wind Speed Problems

When solving wind speed problems, it's important to clearly define the variables and understand the relationship between the speeds involved. The ground speed of an aircraft is influenced by the wind speed, which can either oppose or assist the plane's motion.

Once the variables are established, wind speed problems become a matter of setting up the right equations and solving for the unknowns using algebraic methods. In our example, we visually interpret the given verbal model to set up our equation: plane speed minus wind speed equals ground speed against wind, and plane speed plus wind speed equals ground speed with wind.

Visualizing the scenario helps to grasp the concept that when the plane is going against the wind, the wind speed acts as a negative factor, and when it’s going with the wind, it acts as a positive factor on the ground speed.

Algebraic Problem Solving

Algebraic problem solving requires a systematic approach to handle abstract concepts and variables. We begin by translating the word problem into mathematical expressions, as done with the ground speed calculations.

Clarity is key; therefore, it's vital to define the variables and maintain consistency throughout the problem. The process involves setting up the correct equations based on the relationships mentioned in the problem description.

In our example, the algebraic manipulation used is the method of elimination after the equations are set. We then simplify the resultant equation to solve for one variable and substitute back to find the other. This logical process of manipulating variables and equations is at the heart of algebraic problem solving.

Ground Speed Calculation

Ground speed calculation problems involve determining the speed at which an aircraft travels over the ground. It is a crucial concept as it affects flight time and fuel consumption. In our scenario, the ground speed is affected by wind, and we calculate it by accounting for the plane's speed in still air and wind speed.

The ground speed against the wind is the difference between the plane's speed in still air and the wind speed, while the ground speed with the wind is the sum of these two speeds. This calculation is crucial not only for theoretical exercises but also for practical flight planning and navigation. Pilots and air traffic controllers must consistently consider these calculations for safety and efficiency.