Calculating the Components

Oct 13, 2025 by wbrum.org 121 views

Introduction

In this problem, we will use trigonometry to calculate the distance Amina is from her starting point after walking 50 m on a bearing of 025° and 200 m due east. This problem involves understanding the concept of vectors and how to use them to solve problems in mathematics.

Understanding the Problem

Amina starts at a point and walks 50 m on a bearing of 025°. This means she is walking at an angle of 025° from the north. After that, she walks 200 m due east. We need to find out how far she is from her starting point.

Breaking Down the Problem

To solve this problem, we can break it down into two parts:

Finding the distance Amina walked on a bearing of 025°: We can use the formula for the magnitude of a vector, which is given by:

Magnitude = √(x² + y²)

where x and y are the components of the vector.

In this case, the x-component is 50 m * cos(025°) and the y-component is 50 m * sin(025°).
Finding the distance Amina walked due east: This is simply 200 m.

Calculating the Components

First, let's calculate the x and y components of the vector for the 50 m walk on a bearing of 025°:

x-component = 50 m * cos(025°) = 50 m * 0.9063 = 45.315 m
y-component = 50 m * sin(025°) = 50 m * 0.4226 = 21.13 m

Next, let's calculate the x and y components of the vector for the 200 m walk due east:

x-component = 200 m * cos(0°) = 200 m * 1 = 200 m
y-component = 200 m * sin(0°) = 200 m * 0 = 0 m

Finding the Total Distance

Now, we can add the x and y components of the two vectors to find the total distance Amina walked:

Total x-component = 45.315 m + 200 m = 245.315 m
Total y-component = 21.13 m + 0 m = 21.13 m

Using the Pythagorean Theorem

We can use the Pythagorean theorem to find the distance Amina is from her starting point:

Distance² = (Total x-component)² + (Total y-component)²
Distance² = (245.315 m)² + (21.13 m)²
Distance² = 60451.31 m² + 446.33 m²
Distance² = 60997.64 m²
Distance = √60997.64 m² = 247.93 m

Conclusion

In this problem, we used trigonometry to calculate the distance Amina is from her starting point after walking 50 m on a bearing of 025° and 200 m due east. We broke down the problem into two parts, found the components of the vectors, and used the Pythagorean theorem to find the total distance. The final answer is 247.93 m.

Discussion

This problem involves understanding the concept of vectors and how to use them to solve problems in mathematics. It also requires the use of trigonometry to find the components of the vectors and the Pythagorean theorem to find the total distance.

Solutions

Amina walks 50 m on a bearing of 315° and 200 m due north. The distance Amina is from her starting point is 247.93 m.
Amina walks 50 m on a bearing of 045° and 200 m due east. The distance Amina is from her starting point is 247.93 m.
Amina walks 50 m on a bearing of 225° and 200 m due south. The distance Amina is from her starting point is 247.93 m.

Conclusion

Q: What is the bearing of 025°?

A: The bearing of 025° is an angle of 025° from the north. This means Amina is walking at an angle of 025° from the north.

Q: What is the distance Amina walked on a bearing of 025°?

A: The distance Amina walked on a bearing of 025° is 50 m.

Q: What is the distance Amina walked due east?

A: The distance Amina walked due east is 200 m.

Q: How do we calculate the components of the vectors?

A: To calculate the components of the vectors, we use the formulas:

x-component = distance * cos(angle)
y-component = distance * sin(angle)

Q: What is the x-component of the vector for the 50 m walk on a bearing of 025°?

A: The x-component of the vector for the 50 m walk on a bearing of 025° is 45.315 m.

Q: What is the y-component of the vector for the 50 m walk on a bearing of 025°?

A: The y-component of the vector for the 50 m walk on a bearing of 025° is 21.13 m.

Q: What is the x-component of the vector for the 200 m walk due east?

A: The x-component of the vector for the 200 m walk due east is 200 m.

Q: What is the y-component of the vector for the 200 m walk due east?

A: The y-component of the vector for the 200 m walk due east is 0 m.

Q: How do we find the total distance Amina walked?

A: To find the total distance Amina walked, we add the x and y components of the two vectors:

Total x-component = x-component of 50 m walk + x-component of 200 m walk
Total y-component = y-component of 50 m walk + y-component of 200 m walk

Q: What is the total distance Amina walked?

A: The total distance Amina walked is 247.93 m.

Q: How do we use the Pythagorean theorem to find the distance Amina is from her starting point?

A: To use the Pythagorean theorem to find the distance Amina is from her starting point, we use the formula:

Distance² = (Total x-component)² + (Total y-component)²

Q: What is the distance Amina is from her starting point?

A: The distance Amina is from her starting point is 247.93 m.

Q: What is the final answer?

A: The final answer is 247.93 m.

Conclusion

In this Q&A article, we answered questions related to Amina's journey, including the bearing of 025°, the distance Amina walked on a bearing of 025°, and the distance Amina walked due east. We also explained how to calculate the components of the vectors, find the total distance Amina walked, and use the Pythagorean theorem to find the distance Amina is from her starting point. The final answer is 247.93 m.