https://file.aiquickdraw.com/mathbot/file/1725840829164qqw45l86.jpeg

To solve this problem using the triangle method, we need to break the hiker's displacements into their x (horizontal) and y (vertical) components. We'll sum the components and then use the Pythagorean theorem to find the resultant displacement and its direction.

### Step 1: Break each displacement into components
For each displacement:
- The hiker’s initial walk is **3000 m at 35° north of west**.
- The second displacement is **5500 m at 10° north of west**.
- The third displacement is **1500 m due north**.

#### 1. First displacement (3000 m, 35° north of west):
- **x-component**: \( 3000 \times \cos(35^\circ) \) (westward, so negative)
- **y-component**: \( 3000 \times \sin(35^\circ) \) (northward, so positive)

#### 2. Second displacement (5500 m, 10° north of west):
- **x-component**: \( 5500 \times \cos(10^\circ) \) (westward, so negative)
- **y-component**: \( 5500 \times \sin(10^\circ) \) (northward, so positive)

#### 3. Third displacement (1500 m due north):
- **x-component**: 0 (since it’s due north)
- **y-component**: 1500 (positive, since it's purely northward)

### Step 2: Sum the components
- The total **x-component** of displacement is the sum of all x-components.
- The total **y-component** of displacement is the sum of all y-components.

### Step 3: Calculate the resultant displacement and direction
The resultant displacement is the magnitude of the vector formed by the summed x and y components, calculated using the Pythagorean theorem:

\[
\text{Resultant displacement} = \sqrt{(x_{\text{total}})^2 + (y_{\text{total}})^2}
\]

The direction (θ) of the displacement, relative to west, can be found using the inverse tangent function:

\[
\theta = \tan^{-1}\left(\frac{y_{\text{total}}}{|x_{\text{total}}|}\right)
\]

Let me calculate this now.The hiker's **total displacement** is approximately **8912.66 meters**.

The direction of the resultant displacement is about **27.94° north of west**.

Would you like further details or have any questions?

Here are some related questions for deeper understanding:
1. How does breaking vectors into components help solve displacement problems?
2. What is the significance of using the Pythagorean theorem for vector sums?
3. Why is trigonometry essential in calculating direction in two-dimensional motion?
4. Can you apply this method for displacements at angles other than north and west?
5. How would the result change if the third displacement was not due north?

**Tip:** Always keep track of directions carefully when breaking vectors into components, as the sign (positive or negative) is crucial for accuracy.

A hiker walks from the drop-off and initially covers 3000 m, 35° north of west. He then traveled another 5500 m heading to 10° north of west. Finally, he reached his destination by walking 1500 m due north. What is his total displacement and direction?

Learn how to calculate the resultant displacement and direction for a hiker's journey using vector addition and trigonometry. This problem involves breaking vectors into components, summing them up, and applying the Pythagorean Theorem to find the resultant displacement. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation