https://file.aiquickdraw.com/mathbot/file/17259863373821j28oqsy.jpg

The question asks to determine the magnitude and direction of the resultant displacement for a delivery truck's route. The route consists of:

- A 2.6 km path north.
- A 4.0 km path east.
- A 3.1 km path at a 45° angle (northeast).

### Approach:
1. **Break down the 3.1 km path** into its north and east components using trigonometry:
   - The north component: \(3.1 \times \sin(45^\circ)\).
   - The east component: \(3.1 \times \cos(45^\circ)\).

2. **Add the components** of the route:
   - Total north displacement = \(2.6 + 3.1 \times \sin(45^\circ)\).
   - Total east displacement = \(4.0 + 3.1 \times \cos(45^\circ)\).

3. **Calculate the magnitude** of the resultant displacement using the Pythagorean theorem:
   \[
   \text{Magnitude} = \sqrt{(\text{Total east})^2 + (\text{Total north})^2}
   \]

4. **Find the direction** using the arctangent function:
   \[
   \theta = \tan^{-1}\left(\frac{\text{Total north}}{\text{Total east}}\right)
   \]

I'll calculate these values.The resultant displacement has:

- **Magnitude**: 7.83 km
- **Direction**: 37.74° north of east.

Would you like further details on how the calculations were done or to ask more questions?

Here are 5 related questions for practice:
1. What is the significance of breaking down vectors into components?
2. How does the Pythagorean theorem help in calculating resultant displacement?
3. Why is the angle \( \tan^{-1}(\text{north}/\text{east}) \) used to find direction?
4. How would the result change if the third path had a different angle?
5. Can you calculate the displacement if the truck returned to the starting point?

**Tip**: When dealing with vectors, always break them down into components for easier addition!

A postal employee drives a delivery truck along a path consisting of three segments (2.6 km north, 4.0 km east, and 3.1 km at a 45° angle northeast). Determine the magnitude and direction of the resultant displacement using the component method.

Solve a problem involving a postal delivery route where the truck follows three segments: 2.6 km north, 4.0 km east, and 3.1 km at a 45° angle. Learn how to calculate the resultant displacement using vector addition, trigonometric functions, and the Pythagorean theorem.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation