An Aircraft Travels With The Wind For 120 Miles In 0.75 Of An Hour. The Return Trip Is Flown Against The Wind And Takes Exactly 1 Hour. Which System Of Linear Equations Represents \[$x\$\], The Speed Of The Plane In Miles Per Hour, And
Introduction
In this article, we will explore a real-world scenario involving an aircraft traveling with and against the wind. We will use this scenario to create a system of linear equations that represents the speed of the plane in miles per hour. This problem is a classic example of how algebra can be used to model real-world situations.
The Problem
An aircraft travels with the wind for 120 miles in 0.75 of an hour. The return trip is flown against the wind and takes exactly 1 hour. We are asked to find the system of linear equations that represents the speed of the plane in miles per hour.
Let's Break Down the Problem
Let's denote the speed of the plane as x miles per hour. When the plane travels with the wind, its effective speed is x + y, where y is the speed of the wind. Similarly, when the plane travels against the wind, its effective speed is x - y.
We are given that the plane travels 120 miles in 0.75 of an hour with the wind. We can use the formula distance = speed × time to set up an equation:
120 = (x + y) × 0.75
We can simplify this equation by multiplying both sides by 4:
480 = 4x + 4y
Now, let's consider the return trip. The plane travels against the wind and takes exactly 1 hour to cover the same distance of 120 miles. We can set up another equation using the same formula:
120 = (x - y) × 1
We can simplify this equation by multiplying both sides by 1:
120 = x - y
The System of Linear Equations
We now have two equations:
- 480 = 4x + 4y
- 120 = x - y
We can rewrite these equations in standard form:
- 4x + 4y = 480
- x - y = 120
Solving the System of Linear Equations
We can solve this system of linear equations using either the substitution method or the elimination method. Let's use the elimination method.
First, we can multiply the second equation by 4 to make the coefficients of y in both equations the same:
4(x - y) = 4(120)
This simplifies to:
4x - 4y = 480
Now, we can add this equation to the first equation to eliminate the variable y:
(4x + 4y) + (4x - 4y) = 480 + 480
This simplifies to:
8x = 960
Now, we can divide both sides by 8 to solve for x:
x = 960/8
x = 120
Conclusion
We have found the speed of the plane to be 120 miles per hour. This is the solution to the system of linear equations that represents the speed of the plane in miles per hour.
The Final Answer
The final answer is x = 120.
Discussion
This problem is a great example of how algebra can be used to model real-world situations. The system of linear equations that we created represents the speed of the plane in miles per hour, taking into account the speed of the wind.
Real-World Applications
This problem has many real-world applications, such as:
- Aerodynamics: Understanding the speed of an aircraft in relation to the wind is crucial in aerodynamics.
- Aviation: Pilots need to know the speed of their aircraft in relation to the wind to navigate safely.
- Weather Forecasting: Understanding the speed of the wind is essential in weather forecasting.
Conclusion
Q: What is the speed of the plane in miles per hour?
A: The speed of the plane is 120 miles per hour.
Q: How do we calculate the speed of the plane?
A: We calculate the speed of the plane by using the formula distance = speed × time and taking into account the speed of the wind.
Q: What is the speed of the wind?
A: The speed of the wind is not explicitly given in the problem, but we can find it by substituting the value of x into one of the equations.
Q: How do we find the speed of the wind?
A: We can find the speed of the wind by substituting the value of x into the equation x - y = 120. This gives us:
120 - y = 120
Subtracting 120 from both sides gives us:
-y = 0
Dividing both sides by -1 gives us:
y = 0
So, the speed of the wind is 0 miles per hour.
Q: What if the speed of the wind is not 0? How do we find it?
A: If the speed of the wind is not 0, we can find it by substituting the value of x into one of the equations. Let's use the equation 4x + 4y = 480.
Substituting x = 120 into this equation gives us:
4(120) + 4y = 480
Expanding the left-hand side gives us:
480 + 4y = 480
Subtracting 480 from both sides gives us:
4y = 0
Dividing both sides by 4 gives us:
y = 0
So, the speed of the wind is still 0 miles per hour.
Q: Why is the speed of the wind 0?
A: The speed of the wind is 0 because the problem states that the return trip is flown against the wind and takes exactly 1 hour to cover the same distance of 120 miles. This means that the wind is not affecting the plane's speed.
Q: What if the wind is blowing at a different speed? How do we find the speed of the plane?
A: If the wind is blowing at a different speed, we can find the speed of the plane by using the same method as before. We will need to substitute the value of y into one of the equations and solve for x.
Q: How do we know which equation to use?
A: We can use either equation to find the speed of the plane. However, we need to make sure that we are using the correct value of y.
Q: What if we make a mistake and use the wrong value of y?
A: If we make a mistake and use the wrong value of y, we will get the wrong answer for the speed of the plane. We need to double-check our work and make sure that we are using the correct value of y.
Conclusion
In conclusion, this FAQ article provides answers to some of the most common questions about the problem of finding the speed of the plane. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the problem.