Jack walks 2.4 km on a bearing of 060 degrees followed by a 4.4 km on a bearing 190 degrees. On what bearing and for what distance should he now walk to return directly to his starting point. 


To solve this problem, we can break it down into two steps: calculating the total displacement and then finding the bearing and distance Jack needs to walk to return to his starting point.

### Step 1: Represent the Displacements as Vectors

1. **First leg of the journey:**
   - Distance: 2.4 km
   - Bearing: 060° (measured clockwise from the north)
   - Convert the bearing into a standard angle from the positive x-axis: \(\theta_1 = 90^\circ - 60^\circ = 30^\circ\)
   - Vector components:
     - \(x_1 = 2.4 \cos(30^\circ)\)
     - \(y_1 = 2.4 \sin(30^\circ)\)

2. **Second leg of the journey:**
   - Distance: 4.4 km
   - Bearing: 190° 
   - Convert the bearing into a standard angle from the positive x-axis: \(\theta_2 = 190^\circ - 90^\circ = 100^\circ\)
   - Vector components:
     - \(x_2 = 4.4 \cos(100^\circ)\)
     - \(y_2 = 4.4 \sin(100^\circ)\)

### Step 2: Calculate the Resultant Displacement

Add the x and y components of the two vectors to find the resultant displacement:

\[
x_r = x_1 + x_2
\]
\[
y_r = y_1 + y_2
\]

### Step 3: Calculate the Distance and Bearing for the Return Trip

1. **Distance:**
   \[
   D = \sqrt{x_r^2 + y_r^2}
   \]

2. **Bearing:**
   \[
   \theta_r = \tan^{-1}\left(\frac{y_r}{x_r}\right)
   \]
   Convert \(\theta_r\) back to a bearing from the north.

Let me calculate these for you.Jack should walk approximately **3.40 km** on a bearing of **157.24 degrees** to return directly to his starting point.

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. How can you use the sine and cosine rules in non-right-angled triangle problems?
2. What is the significance of using bearings in navigation?
3. How do you convert between Cartesian coordinates and polar coordinates?
4. Can you determine the area of a triangle given two sides and the included angle?
5. How do you calculate the distance between two points in a coordinate plane?

**Tip:** When dealing with bearings, always remember that they are measured clockwise from the north, and converting them into standard angles is key in vector calculations.

Jack walks 2.4 km on a bearing of 060 degrees followed by a 4.4 km on a bearing 190 degrees. On what bearing and for what distance should he now walk to return directly to his starting point?

Discover how to calculate the distance and bearing Jack needs to walk to return to his starting point after walking two legs with different bearings and distances. This problem involves trigonometry, vector addition, and bearings, making it suitable for students in Grades 10-12.

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