Question

Computing Wind Speed The average airspeed of a single-engine aircraft is 150 miles per hour. If the aircraft flew the same distance in 2 hours with the wind as it flew in 3 hours against the wind, what was the wind speed?

Short answer

The wind speed is 30 miles per hour.

Step-by-step solution

- Define Variables

Let the wind speed be denoted as \( w \) miles per hour. The airspeed of the aircraft is given as 150 miles per hour.

- Equation for With the Wind

When flying with the wind, the speed of the aircraft is \( 150 + w \) miles per hour. The time taken is 2 hours. Thus, the distance traveled can be expressed as: \[(150 + w) \times 2\].

- Equation for Against the Wind

When flying against the wind, the speed of the aircraft is \( 150 - w \) miles per hour. The time taken is 3 hours. Thus, the distance traveled can be expressed as: \[(150 - w) \times 3\].

- Set Up Equation

Since the distances are the same, set the two equations equal to each other and solve for \( w \): \[(150 + w) \times 2 = (150 - w) \times 3\].

- Simplify and Solve

Expand both sides of the equation: \[(150 + w) \times 2 = 300 + 2w\] \[(150 - w) \times 3 = 450 - 3w\]. Now set the equations equal to each other: \ 300 + 2w = 450 - 3w \. Combine like terms: \ 5w = 150 \. Solve for \( w \): \ w = 30 \.

Key concepts

distance-speed-time relationship

Understanding the distance-speed-time relationship is fundamental in solving problems involving motion. The key formula to remember here is:
Distance = Speed × Time.
This equation links how far an object travels with how fast it is going and for how long.
Let's break it down:

• Distance is how far something travels, typically measured in miles, kilometers, etc.
• Speed is the rate at which the object travels, usually measured in miles per hour (mph), kilometers per hour (km/h), etc.
• Time is the duration of the travel, measured in hours, minutes, etc.

In this problem, the aircraft's airspeed is given as 150 mph. When flying with the wind, the wind speed adds to the aircraft's speed, while against the wind, it slows it down.

algebraic equations

Algebraic equations are used to express relationships between variables.
In our problem, we define the variable w as the wind speed.
This helps us create equations that represent the different scenarios of the flight.
Here are the equations formed:

• With the wind: (150 + w) × 2
• Against the wind: (150 - w) × 3

These equations describe the distances traveled under each condition.
Because the aircraft travels the same distance in both cases, we set these two equations equal to one another.
Algebraic equations allow us to model real-world problems and solve for unknown values.

solving linear equations

Solving linear equations often involves isolating the unknown variable. In our problem, we set up the equation (150 + w) × 2 = (150 - w) × 3.
Let's simplify this step-by-step:

• Expand both sides: 300 + 2w = 450 - 3w
• Combine like terms: Bring all terms involving w on one side: 300 + 2w + 3w = 450
• Simplify: 300 + 5w = 450
• Solve for w by isolating it: 5w = 150 → w = 30

This means the wind speed is 30 mph.
Linear equations form the basis for many algebra problems, and understanding how to manipulate and solve them is crucial.
This process involves expanding, combining like terms, and isolating variables.