Problem 46 - Algebra 1

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 Solve the linear system.

Step-by-Step Solution

Answer

The speed of the plane in still air is 375 mph, and the wind speed is 75 mph.

Step 1: Identify the equations

From the information stated and the verbal model provided, two equations can be formulated:\n1) Speed in still air - Wind speed = Ground speed against wind \n2) Speed in still air + Wind speed = Ground speed with wind \nSubstitute the known ground speeds from the problem into these formulas to get:\n1) Speed in still air - Wind speed = 300 (against the wind) \n2) Speed in still air + Wind speed = 450 (with the wind)

Step 2: Process for Solving the Equations

To find the speed in still air and the wind speed, these two equations can be solved simultaneously. There are multiple ways to solve a system of linear equations - substitution, addition (elimination), or graphical methods. In this case, addition (elimination) is the easiest method. By adding the two equations together, the 'Wind speed' variable is eliminated, enabling the calculation of the 'Speed in still air'.

Step 3: Solve for Speed in Still Air

Add the equations: \n(Speed in still air - Wind speed) + (Speed in still air + Wind speed) = 300 + 450 \nThis simplifies to: 2 * Speed in still air = 750 \nSo Speed in still air = 750 / 2 = 375 mph.

Step 4: Solve for Wind Speed

Substitute the 'Speed in still air' into the first equation to find the 'Wind speed': \n375 - Wind speed = 300 \nSo Wind speed = 375 - 300 = 75 mph.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that express the equality of two expressions. They often involve variables and constants, connected by operations like addition, subtraction, multiplication, and division.
In the context of solving problems related to linear systems, algebraic equations are crucial. They help us represent real-world situations in mathematical form. This allows us to find unknown values systematically.
In the exercise, two algebraic equations are set up using information about a plane's speed and the wind's influence:

• The speed of the plane without wind (let's call it "Speed in still air") is unknown, but we need to calculate it.
• The ground speed against the wind is given as 300 mph, so we have: (Speed in still air) - (Wind Speed) = 300.
• Conversely, the ground speed with the wind is 450 mph, giving us: (Speed in still air) + (Wind Speed) = 450.

These equations form a system of linear equations that can be solved using methods like substitution or elimination, as outlined in the step-by-step solution.

Wind Speed Calculations

Calculating wind speed is essential to understanding how it affects travel, especially for airplanes. It can either slow down or speed up a plane, depending on the direction of the wind relative to the flight.
In the provided exercise, a verbal model helps conceptualize how wind affects the plane's speed:

• With wind: The ground speed is higher because the wind adds to the plane's actual speed.
• Against wind: The ground speed is lower as the wind subtracts from the plane's speed.

These concepts help in forming the equations needed for solving the system. Once we know the ground speeds both with and against the wind, we can calculate the wind speed by solving the equations.
Using the addition method, after determining the speed of the plane in still air to be 375 mph, the wind speed can be calculated by solving: 375 - Wind speed = 300,
resulting in Wind speed = 375 - 300 = 75 mph.

Ground Speed

Ground speed is an important concept in aviation. It refers to the speed of an aircraft relative to the ground. It is influenced by factors such as wind speed and direction.
In our exercise, two ground speeds are given: against the wind (300 mph) and with the wind (450 mph). These define how fast the plane moves across the earth's surface, accounting for wind effects.

• Ground speed against the wind reflects the airplane's struggle against wind resistance and is thus slower.
• Ground speed with the wind showcases how the plane joyfully rides along with the wind, resulting in a higher speed.

Understanding ground speed is crucial for flight planning and fuel calculations. It ensures that arrival times and routes are accurately determined, accounting for external influences on the aircraft's performance. With both forms of ground speed provided, we can effectively deduce both the speed of the plane in still air and the wind speed through a systematic approach with algebraic equations.