A postal employee drives a delivery truck over the route shown in A route is marked on a map. The route begins with a 2.6-kilometer-long segment going to the north from the starting point. Then it goes for 4.0 kilometers to the east. The last segment goes for 3.1 kilometers in the north-east direction, 45 degrees east of north, to the stop point..

Use the method of components to determine the magnitude of her resultant displacement.
Express your answer in kilometers.

displacement| = Km

To solve this problem using the method of components, we need to break each segment of the postal employee's route into its north and east components and then sum these components to find the resultant displacement.

### Step-by-Step Solution

1. **Break down each segment into components**:

   - **First segment (2.6 km north)**:
     - North component: \(2.6\) km
     - East component: \(0\) km
   
   - **Second segment (4.0 km east)**:
     - North component: \(0\) km
     - East component: \(4.0\) km
   
   - **Third segment (3.1 km at 45 degrees northeast)**:
     - Since the segment is at a 45-degree angle to both north and east, we use trigonometric functions to find the components:
     \[
     \text{North component} = 3.1 \times \cos(45^\circ)
     \]
     \[
     \text{East component} = 3.1 \times \sin(45^\circ)
     \]
     Since \(\cos(45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.707\):
     \[
     \text{North component} = 3.1 \times 0.707 \approx 2.192 \, \text{km}
     \]
     \[
     \text{East component} = 3.1 \times 0.707 \approx 2.192 \, \text{km}
     \]

2. **Calculate the total north and east components**:

   \[
   \text{Total North component} = 2.6 + 0 + 2.192 = 4.792 \, \text{km}
   \]
   \[
   \text{Total East component} = 0 + 4.0 + 2.192 = 6.192 \, \text{km}
   \]

3. **Determine the magnitude of the resultant displacement**:

   The magnitude of the resultant displacement \(|\text{displacement}|\) can be found using the Pythagorean theorem:
   \[
   |\text{displacement}| = \sqrt{(\text{Total North component})^2 + (\text{Total East component})^2}
   \]
   \[
   |\text{displacement}| = \sqrt{(4.792)^2 + (6.192)^2}
   \]
   \[
   |\text{displacement}| = \sqrt{22.970 + 38.338}
   \]
   \[
   |\text{displacement}| = \sqrt{61.308}
   \]
   \[
   |\text{displacement}| \approx 7.83 \, \text{km}
   \]

### Result

The magnitude of the postal employee's resultant displacement is approximately **7.83 kilometers**.

Would you like more details or have any questions about this solution?

Here are some related questions for further exploration:

1. How would the displacement change if the second segment were longer?
2. What would be the total distance traveled by the postal employee?
3. How would you find the direction of the resultant displacement relative to the north?
4. What happens to the components if the third segment was in a different direction?
5. If the angles of segments changed, how would that affect the displacement calculation?

**Tip:** When solving vector problems, always start by breaking down each vector into its components. This simplifies the calculations and allows you to easily find the resultant vector.

Learn how to calculate the magnitude of resultant displacement using vector components and trigonometric functions. This problem involves breaking down vectors into north and east components, applying trigonometric functions to find components at an angle, and using the Pythagorean theorem to determine magnitude. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation