Plane Overtake: Step-by-Step Math Solution

Aug 6, 2025 by Luna Greco 43 views

When Will the Faster Plane Overtake the Slower One? A Step-by-Step Solution

Have you ever wondered about the math behind airplanes overtaking each other? It's a classic problem that combines distance, rate, and time. Let's break down a real-world scenario and solve it together! This comprehensive guide will walk you through each step, making it easy to understand. This article will dissect a classic problem of relative speed and time, providing a clear, step-by-step solution that anyone can follow. So, buckle up, and let's dive into the world of aviation mathematics!

The Problem: A Race in the Sky

Okay, guys, here's the situation: Imagine two airplanes taking off from Atlanta. The first airplane departs at 2 PM, heading north at a steady 250 miles per hour. Half an hour later, at 2:30 PM, a second airplane leaves, flying north at a faster speed of 280 miles per hour. The big question is: At what time will the second airplane catch up to the first one? This isn't just a theoretical problem; it's the kind of calculation air traffic controllers and pilots deal with regularly. So, understanding how to solve it is pretty crucial.

Breaking Down the Problem

Before we jump into the calculations, let's break down what we know and what we need to find out. We know the departure times of both planes, their speeds, and the direction they're traveling. What we need to figure out is the time when the second plane overtakes the first. To solve this, we'll need to use the fundamental relationship between distance, rate, and time, which is distance = rate × time. We'll also need to consider the head start that the first plane has. This problem involves understanding the concept of relative speed, which is how much faster one object is moving compared to another. In this case, it's the difference in speed between the two planes. The key to solving this problem lies in understanding how the head start affects the time it takes for the faster plane to catch up. We will meticulously analyze the distances covered by each plane at different times, ensuring a clear understanding of the problem's dynamics. So, let's gather our facts and figures and prepare for a mathematical journey through the skies!

Step 1: Calculate the Head Start

First things first, let's figure out how much of a head start the first airplane has. It takes off at 2 PM, and the second plane leaves 30 minutes later. So, the first plane has a 30-minute head start. To make our calculations easier, let's convert that 30 minutes into hours. Since there are 60 minutes in an hour, 30 minutes is equal to 0.5 hours. Now, let's calculate how far the first plane travels in those 0.5 hours. We know its speed is 250 miles per hour, so we use our trusty formula: distance = rate × time.

Distance = 250 miles/hour × 0.5 hours = 125 miles

So, the first plane has a 125-mile head start. This initial distance is crucial for our calculations. Understanding the head start is the first key to unlocking the solution. This 125-mile gap is what the second plane needs to close, and it will do so at a rate determined by the difference in their speeds. This step lays the groundwork for understanding the relative speeds and the time it will take for the second plane to catch up. This initial calculation sets the stage for the rest of the solution, giving us a concrete value to work with. Now that we have the head start distance, we can move on to the next step: calculating the relative speed. This will tell us how quickly the second plane is closing the gap.

Step 2: Determine the Relative Speed

Now, let's figure out how much faster the second plane is compared to the first. This is what we call the relative speed. We know the second plane is flying at 280 miles per hour, and the first plane is flying at 250 miles per hour. To find the relative speed, we simply subtract the slower speed from the faster speed.

Relative Speed = 280 miles/hour - 250 miles/hour = 30 miles/hour

So, the second plane is closing the distance at a rate of 30 miles per hour. This relative speed is the rate at which the second plane is gaining on the first. Understanding this concept is essential for solving problems involving moving objects. This 30 mph difference is the key to understanding how quickly the second plane will close the 125-mile gap. This step is vital because it simplifies the problem to one of catching up, rather than calculating the individual distances of each plane. With the relative speed determined, we can now calculate the time it will take for the second plane to overtake the first. This is where the problem starts to come together, and we can see the light at the end of the mathematical tunnel.

Step 3: Calculate the Time to Overtake

We're getting closer! We know the first plane has a 125-mile head start, and the second plane is closing the gap at 30 miles per hour. Now, we can calculate the time it will take for the second plane to overtake the first. We'll use the same formula again: time = distance / rate. In this case, the distance is the head start (125 miles), and the rate is the relative speed (30 miles per hour).

Time = 125 miles / 30 miles/hour = 4.1667 hours

So, it will take the second plane approximately 4.1667 hours to overtake the first. This time calculation is the heart of the problem. Understanding how to apply the formula time = distance / rate is crucial here. This 4.1667 hours is the key to determining the exact time of the overtake. But what does 4.1667 hours mean in terms of hours and minutes? We need to convert the decimal part of the hours into minutes to get a more precise time. This conversion will give us a clearer picture of when the second plane will catch up. Let's move on to the final step of converting this time into a more understandable format.

Step 4: Convert to Hours and Minutes

Okay, we've got 4.1667 hours, but that's not exactly how we tell time, right? We need to break that down into hours and minutes. We already have the 4 full hours, so let's focus on the 0.1667 part. To convert this decimal into minutes, we multiply it by 60 (since there are 60 minutes in an hour).

Minutes = 0.1667 hours × 60 minutes/hour ≈ 10 minutes

So, 0.1667 hours is approximately equal to 10 minutes. That means it will take the second plane 4 hours and 10 minutes to overtake the first. This conversion is essential for providing the answer in a practical format. Understanding how to convert decimal hours to minutes is a handy skill. These 10 minutes make our time calculation more precise and easier to understand in real-world terms. Now that we have the time in hours and minutes, we can finally determine the exact time when the second plane will overtake the first. This is the moment we've been working towards, and it's the final piece of the puzzle.

Step 5: Determine the Overtake Time

We're in the home stretch! We know the second plane leaves at 2:30 PM, and it takes 4 hours and 10 minutes to overtake the first plane. To find the overtake time, we simply add 4 hours and 10 minutes to 2:30 PM.

2:30 PM + 4 hours = 6:30 PM
6:30 PM + 10 minutes = 6:40 PM

Therefore, the second airplane will overtake the first at 6:40 PM. This final calculation gives us the answer we've been seeking. Understanding how to add time is a practical skill that we use every day. 6:40 PM is the time when the second plane will finally catch up to the first, completing our calculation. This is the culmination of all our previous steps, and it provides a clear and definitive answer to the problem. We've successfully navigated the complexities of relative speed and time, and we've arrived at our destination: the time of the overtake. Congratulations, math explorers!

Answer

The correct answer is C. 6:40 PM. This solution demonstrates the power of breaking down complex problems into smaller, manageable steps. Understanding the concepts of relative speed and time is essential for solving these types of problems. This 6:40 PM mark is the result of our careful calculations and step-by-step approach. We've successfully solved a real-world problem using mathematical principles, and that's something to be proud of. Now, let's recap the key steps we took to reach this solution.

Summary of the Solution

Let's quickly recap how we solved this problem, guys:

We calculated the head start of the first plane: 125 miles.
We determined the relative speed between the two planes: 30 miles per hour.
We calculated the time it would take for the second plane to overtake the first: 4.1667 hours.
We converted the time into hours and minutes: 4 hours and 10 minutes.
Finally, we determined the overtake time: 6:40 PM.

These steps provide a clear roadmap for solving similar problems in the future. Understanding the process is just as important as getting the right answer. This summary serves as a quick reference guide, allowing you to revisit the key steps whenever needed. We've not only solved the problem but also learned a valuable problem-solving strategy that can be applied in various situations. Remember, breaking down a complex problem into smaller, manageable steps makes it much easier to tackle. Keep these steps in mind, and you'll be well-equipped to solve similar challenges in the future.

Practice Makes Perfect

Now that you've seen how to solve this problem, why not try some similar ones? You can change the speeds, departure times, or even the direction of the planes. The key is to practice and get comfortable with the concepts. Remember, practice makes perfect. Try varying the parameters of the problem to test your understanding. Experimenting with different scenarios will solidify your grasp of the concepts. The more you practice, the more confident you'll become in your problem-solving abilities. So, grab a pencil and paper, and start exploring the world of aviation mathematics! There are countless variations of this problem that you can try, and each one will help you hone your skills. Remember, the journey of learning is just as important as the destination. So, embrace the challenge and enjoy the process of discovery.

Conclusion

So, there you have it! We've successfully solved a classic problem involving relative speed and time. By breaking down the problem into smaller steps and using the fundamental relationship between distance, rate, and time, we were able to find the exact time when the second plane would overtake the first. I hope this step-by-step guide has been helpful and has demystified this type of problem for you. Remember to practice, stay curious, and keep exploring the world of mathematics! This problem-solving approach can be applied to various real-world scenarios. Understanding the principles behind this solution will empower you to tackle other challenges. We've not only solved a problem but also gained a deeper understanding of the concepts involved. Keep this knowledge with you, and you'll be well-prepared to navigate the world of mathematics with confidence. And who knows, maybe you'll even become a pilot or an air traffic controller someday!