Calculating the Net Displacement

Oct 13, 2025 by miodeya.com 70 views

Introduction

In this article, we will delve into the world of mathematics and explore a fascinating problem involving a path walked by Felipe. The problem is presented in a simple yet intriguing way, and we will use mathematical concepts to determine the coordinates where Felipe ends his walk. We will break down the problem step by step, using geometric and algebraic techniques to arrive at the solution.

The Problem

Felipe starts his walk at point (-4, 1) and follows a specific pattern:

Walk 1 meter and turn left.
Walk 2 meters and turn left.
Walk 3 meters and turn left.

He repeats this pattern 7 times. Our goal is to find the coordinates where Felipe ends his walk.

Breaking Down the Pattern

Let's analyze the pattern and understand what happens at each step. When Felipe walks 1 meter and turns left, he moves in a counterclockwise direction. This means that his x-coordinate decreases by 1, and his y-coordinate increases by 1.

Similarly, when he walks 2 meters and turns left, his x-coordinate decreases by 2, and his y-coordinate increases by 2. Finally, when he walks 3 meters and turns left, his x-coordinate decreases by 3, and his y-coordinate increases by 3.

Calculating the Net Displacement

Since Felipe repeats the pattern 7 times, we need to calculate the net displacement after each repetition. Let's calculate the net displacement for each step:

After the first repetition, Felipe's x-coordinate decreases by 1 + 2 + 3 = 6, and his y-coordinate increases by 1 + 2 + 3 = 6.
After the second repetition, Felipe's x-coordinate decreases by another 6 (since the pattern repeats), and his y-coordinate increases by another 6.
We can continue this process for each repetition, but it's easier to use a geometric approach.

Geometric Approach

Let's visualize Felipe's path as a series of connected line segments. Each line segment represents a single repetition of the pattern. We can see that the x-coordinates form an arithmetic sequence with a common difference of -6, and the y-coordinates form an arithmetic sequence with a common difference of 6.

Finding the Final Coordinates

To find the final coordinates, we need to calculate the net displacement after 7 repetitions. We can use the formula for the sum of an arithmetic series:

For the x-coordinates: x = -6(1 + 2 + ... + 7)
For the y-coordinates: y = 6(1 + 2 + ... + 7)

Using the formula for the sum of an arithmetic series, we get:

x = -6(28) = -168
y = 6(28) = 168

Therefore, the final coordinates where Felipe ends his walk are (-168, 168).

Conclusion

In this article, we explored a fascinating problem involving a path walked by Felipe. We used geometric and algebraic techniques to determine the coordinates where Felipe ends his walk. By breaking down the pattern and calculating the net displacement, we arrived at the solution. This problem is a great example of how mathematical concepts can be applied to real-world problems, and we hope that this article has provided a clear and concise explanation of the solution.

Additional Resources

For those who want to explore more mathematical problems, we recommend checking out the following resources:

Math Open Reference: A comprehensive online reference for mathematics.
Khan Academy: A free online platform for learning mathematics and other subjects.
MIT OpenCourseWare: A free online resource for mathematics and other subjects.

Introduction

In our previous article, we explored a fascinating problem involving a path walked by Felipe. We used geometric and algebraic techniques to determine the coordinates where Felipe ends his walk. In this article, we will answer some of the most frequently asked questions about Felipe's path.

Q&A

Q: What is the pattern that Felipe follows?

A: Felipe follows a pattern where he walks 1 meter and turns left, then walks 2 meters and turns left, and finally walks 3 meters and turns left. He repeats this pattern 7 times.

Q: How do we calculate the net displacement after each repetition?

A: We can calculate the net displacement by adding up the x-coordinates and y-coordinates separately. For the x-coordinates, we subtract 1 + 2 + 3 = 6 for each repetition. For the y-coordinates, we add 1 + 2 + 3 = 6 for each repetition.

Q: Can we use a geometric approach to solve the problem?

A: Yes, we can use a geometric approach to solve the problem. We can visualize Felipe's path as a series of connected line segments, where each line segment represents a single repetition of the pattern.

Q: How do we find the final coordinates where Felipe ends his walk?

A: To find the final coordinates, we need to calculate the net displacement after 7 repetitions. We can use the formula for the sum of an arithmetic series to calculate the net displacement.

Q: What are the final coordinates where Felipe ends his walk?

A: The final coordinates where Felipe ends his walk are (-168, 168).

Q: Can we use this problem to teach other mathematical concepts?

A: Yes, this problem can be used to teach other mathematical concepts such as arithmetic sequences, geometric sequences, and coordinate geometry.

Q: How can we modify the problem to make it more challenging?

A: We can modify the problem by changing the pattern that Felipe follows, or by increasing the number of repetitions. We can also add additional constraints, such as limiting the distance that Felipe can walk.

Q: Can we use this problem to teach problem-solving skills?

A: Yes, this problem can be used to teach problem-solving skills such as breaking down complex problems into smaller parts, identifying patterns, and using mathematical concepts to solve the problem.

Conclusion

In this article, we answered some of the most frequently asked questions about Felipe's path. We hope that this article has provided a useful and engaging exploration of the problem, and we encourage readers to try out the solution for themselves.

Additional Resources

For those who want to explore more mathematical problems, we recommend checking out the following resources:

Math Open Reference: A comprehensive online reference for mathematics.
Khan Academy: A free online platform for learning mathematics and other subjects.
MIT OpenCourseWare: A free online resource for mathematics and other subjects.

We hope that this article has provided a useful and engaging exploration of the problem, and we encourage readers to try out the solution for themselves.