Two racing boats set out from the same dock and speed away at the same constant speed of 119 km/h for half an hour (0.500 h), the blue boat headed 26.8° south of west, and the green boat headed 41.4° south of west. During this half-hour (a) how much farther west does the blue boat travel, compared to the green boat

To solve this problem, we need to break down the motion of the boats into their northward and westward components.

Let's start by finding the westward component for each boat. We can use trigonometry to calculate it.

For the blue boat:
The direction of 26.8° south of west can be represented as 180° - 26.8° = 153.2° west of north.
The westward component is then calculated by multiplying the speed (119 km/h) by the cosine of the angle:
westward component for blue boat = 119 km/h * cos(153.2°)

For the green boat:
The direction of 41.4° south of west can be represented as 180° - 41.4° = 138.6° west of north.
The westward component is then calculated by multiplying the speed (119 km/h) by the cosine of the angle:
westward component for green boat = 119 km/h * cos(138.6°)

Now that we have the westward components for both boats, we can compare them.

westward distance traveled by blue boat = westward component for blue boat * time (0.5 hours)
westward distance traveled by green boat = westward component for green boat * time (0.5 hours)

Simply subtracting the westward distance traveled by the green boat from the westward distance traveled by the blue boat will give us the answer.

westward distance difference = westward distance traveled by blue boat - westward distance traveled by green boat

To summarize, here are the steps to solve the problem:

1. Convert the directions of the boats to angles west of north.
2. Calculate the westward components of each boat using the speed and the cosine of the angles.
3. Multiply the westward components by the time (0.5 hours).
4. Subtract the westward distance traveled by the green boat from the westward distance traveled by the blue boat to find the difference.