To determine the bearing of the car's movement, we need to calculate the angle between the direction the car traveled and the north direction. The bearing is measured clockwise from the north direction, typically expressed in degrees.

In this case:

• The car ends up 30 km east and 75 km south of its starting point.

We can visualize this situation as a right-angled triangle where:

• The horizontal leg represents the eastward displacement (30 km),
• The vertical leg represents the southward displacement (75 km).

To calculate the bearing, we first need to find the angle $\theta$ relative to the east-west axis using the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{75}{30}$

Now, we can calculate the angle $\theta$, which is the angle south of the eastward direction.

$\theta = \tan^{-1}\left(\frac{75}{30}\right)$

Let me calculate that.The angle $\theta$ south of the eastward direction is approximately 68.2°.

To find the bearing, which is measured clockwise from north, we add this angle to 90°, since the east direction is 90° from north. Therefore, the bearing is:

$\text{Bearing} = 90^\circ + 68.2^\circ = 158.2^\circ$

Thus, the bearing on which the car traveled is approximately 158.2°.

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. How do we find bearings in situations involving other directions, like west or north?
2. What is the significance of bearings in navigation and positioning?
3. How does the tangent function help in solving right-angled triangle problems like this?
4. Can you explain how to convert between compass bearings and angles from standard position?
5. How can GPS systems calculate the bearing between two points on Earth?

Tip: Bearings always start from the north direction, moving clockwise, and they can be used in navigation, mapping, and surveying to determine direction accurately.