Hilary and Dale leave camp and go for a long hike. After going 7 km due east, they

turn and go another 8 km in the direction 60 degrees north of east. They plan to return along a

straight path. How far from camp are they at this point? Use an angle to describe the direction

that Hilary and Dale should follow to reach their camp.

To solve this problem, we can treat it as a vector problem and break it into two parts.

### Step 1: Represent the movement in vectors
1. **First leg (7 km due east)**:
   - This leg is purely in the east direction, so we represent it as a vector:
   \[
   \vec{A} = (7, 0)
   \]
   This means they travel 7 km along the x-axis (east) and 0 km along the y-axis (north).

2. **Second leg (8 km at 60° north of east)**:
   - The second leg is at a 60° angle north of east. To break it into x and y components:
   \[
   \text{x-component of } \vec{B} = 8 \cos(60^\circ) = 8 \times \frac{1}{2} = 4
   \]
   \[
   \text{y-component of } \vec{B} = 8 \sin(60^\circ) = 8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3} \approx 6.93
   \]
   So the vector for the second leg is:
   \[
   \vec{B} = (4, 6.93)
   \]

### Step 2: Find the total displacement
Now, let's add the vectors \(\vec{A}\) and \(\vec{B}\) to find the total displacement from the camp:
\[
\vec{A} + \vec{B} = (7 + 4, 0 + 6.93) = (11, 6.93)
\]

### Step 3: Calculate the straight-line distance from the camp
The straight-line distance is the magnitude of the resultant displacement vector:
\[
d = \sqrt{(11)^2 + (6.93)^2} = \sqrt{121 + 48.04} = \sqrt{169.04} \approx 13 km
\]
So, they are approximately **13 km** from camp.

### Step 4: Find the angle of their return path
To describe the direction, we need to find the angle \(\theta\) relative to due east. The angle can be found using the tangent function:
\[
\tan \theta = \frac{\text{y-component}}{\text{x-component}} = \frac{6.93}{11}
\]
\[
\theta = \tan^{-1}\left(\frac{6.93}{11}\right) \approx 32.88^\circ
\]
So, they should travel **32.88° south of west** to return to camp.

### Final Answer:
- Distance from camp: **13 km**
- Direction to return: **32.88° south of west**

Would you like more details on any part of this process? 

Here are some related questions to consider:
1. What would happen if the second leg had a different angle?
2. How would the distance and angle change if the hike had been longer?
3. What if they had traveled west first instead of east?
4. How would we calculate the return angle if they hiked only north and south?
5. How does changing the angle of the second leg affect the return path?

**Tip**: Breaking vectors into components using trigonometry simplifies many navigation and physics problems!

Hilary and Dale leave camp and go for a long hike. After going 7 km due east, they turn and go another 8 km in the direction 60 degrees north of east. They plan to return along a straight path. How far from camp are they at this point? Use an angle to describe the direction that Hilary and Dale should follow to reach their camp.

Solve a hiking problem involving vector addition and trigonometry. Hilary and Dale travel 7 km east and 8 km at 60° north of east. Find the distance from camp (13 km) and the angle of return (32.88° south of west). Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation