To determine how far the boy is from his starting point, let's follow a step-by-step solution:
1. Initial Displacement:
- The boy initially walks 4 km due west. This means his initial movement is 4 km along the negative x-axis.
2. Angle and Bearing:
- Next, the boy changes direction to walk on a bearing of 214°. Bearing is measured clockwise from the north direction.
- To convert this bearing to an angle measured from the positive x-axis:
[tex]\[
214^\circ - 180^\circ = 34^\circ
\][/tex]
This converted angle of 34° is measured counterclockwise from the negative x-axis.
3. Vector Components:
- We now need to determine the components of the displacement vector from the initial point after changing direction.
- The x-component (west direction) can be found using the cosine of the angle:
[tex]\[
x_2 = 4 \cdot \cos(34^\circ) \approx 3.316
\][/tex]
- The y-component (south direction) can be found using the sine of the angle:
[tex]\[
y_2 = 4 \cdot \sin(34^\circ) \approx 2.237
\][/tex]
4. Distance Calculation:
- Finally, we use the Pythagorean theorem to calculate the total distance from the starting point. Since the components form a right triangle with the hypotenuse being the resultant displacement:
[tex]\[
\text{Distance} = \sqrt{x_2^2 + y_2^2}
\approx \sqrt{(3.316)^2 + (2.237)^2}
\approx 4.0 \text{ km}
\][/tex]
Therefore, the boy is approximately 4.0 km from his starting point.
