Two racing boats set out from the same dock and speed away at the same constant speed of 102 km/h for half an hour (0.500 h), the blue boat headed 26.1° south of west, and the green boat headed 38.8° south of west. During this half-hour (a) how much farther west does the blue boat travel, compared to the green boat, and (b) how much farther south does the green boat travel, compared to the blue boat? Express your answers in km

To find the answers, we can break down the velocities of the boats into horizontal (westward) and vertical (southward) components.

For the blue boat:
Velocity in the westward direction = 102 km/h * cos(26.1°)
Velocity in the southward direction = 102 km/h * sin(26.1°)

For the green boat:
Velocity in the westward direction = 102 km/h * cos(38.8°)
Velocity in the southward direction = 102 km/h * sin(38.8°)

(a) The distance traveled westward by the blue boat compared to the green boat can be found by subtracting the westward components:
Distance traveled westward by the blue boat = (102 km/h * cos(26.1°)) * (0.500 h)
Distance traveled westward by the green boat = (102 km/h * cos(38.8°)) * (0.500 h)
Difference in the distance traveled westward = Distance traveled westward by the blue boat - Distance traveled westward by the green boat

(b) The distance traveled southward by the green boat compared to the blue boat can be found by subtracting the southward components:
Distance traveled southward by the blue boat = (102 km/h * sin(26.1°)) * (0.500 h)
Distance traveled southward by the green boat = (102 km/h * sin(38.8°)) * (0.500 h)
Difference in the distance traveled southward = Distance traveled southward by the green boat - Distance traveled southward by the blue boat

Let's calculate each value step by step.

Velocity in the westward direction for the blue boat:
Velocity_blue_west = 102 km/h * cos(26.1°)
Velocity_blue_west = 92.0105 km/h

Velocity in the southward direction for the blue boat:
Velocity_blue_south = 102 km/h * sin(26.1°)
Velocity_blue_south = 44.6855 km/h

Velocity in the westward direction for the green boat:
Velocity_green_west = 102 km/h * cos(38.8°)
Velocity_green_west = 80.2747 km/h

Velocity in the southward direction for the green boat:
Velocity_green_south = 102 km/h * sin(38.8°)
Velocity_green_south = 63.9367 km/h

(a) Distance traveled westward by the blue boat:
Distance_blue_west = Velocity_blue_west * (0.500 h)
Distance_blue_west = 46.0053 km

Distance traveled westward by the green boat:
Distance_green_west = Velocity_green_west * (0.500 h)
Distance_green_west = 40.1374 km

Difference in the distance traveled westward:
Difference_west = Distance_blue_west - Distance_green_west
Difference_west = 5.8679 km

(b) Distance traveled southward by the green boat:
Distance_green_south = Velocity_green_south * (0.500 h)
Distance_green_south = 31.96835 km

Distance traveled southward by the blue boat:
Distance_blue_south = Velocity_blue_south * (0.500 h)
Distance_blue_south = 22.34275 km

Difference in the distance traveled southward:
Difference_south = Distance_green_south - Distance_blue_south
Difference_south = 9.6256 km

Therefore, the answers are:
(a) The blue boat travels 5.8679 km farther west than the green boat.
(b) The green boat travels 9.6256 km farther south than the blue boat.

To find the answers, we need to calculate the distances traveled by each boat in the west and south directions.

First, we find the total distance traveled by both boats in half an hour:

Distance = Speed x Time
Distance = 102 km/h x 0.500 h
Distance = 51 km

(a) To calculate how much farther west the blue boat travels compared to the green boat, we need to find the westward components of their distances.

For the blue boat:
Westward Distance (blue) = Distance x cos(angle)
Westward Distance (blue) = 51 km x cos(26.1°)

For the green boat:
Westward Distance (green) = Distance x cos(angle)
Westward Distance (green) = 51 km x cos(38.8°)

(b) To calculate how much farther south the green boat travels compared to the blue boat, we need to find the southward components of their distances.

For the blue boat:
Southward Distance (blue) = Distance x sin(angle)
Southward Distance (blue) = 51 km x sin(26.1°)

For the green boat:
Southward Distance (green) = Distance x sin(angle)
Southward Distance (green) = 51 km x sin(38.8°)

Now, let's calculate the values:

(a) Westward Distance (blue) = 51 km x cos(26.1°)
(b) Westward Distance (green) = 51 km x cos(38.8°)
(a) Farther West = Westward Distance (blue) - Westward Distance (green)

(a) Farther West = 51 km x cos(26.1°) - 51 km x cos(38.8°)

(b) Southward Distance (blue) = 51 km x sin(26.1°)
(b) Southward Distance (green) = 51 km x sin(38.8°)
(b) Farther South = Southward Distance (green) - Southward Distance (blue)

(b) Farther South = 51 km x sin(38.8°) - 51 km x sin(26.1°)

Now, we can calculate the values using a scientific calculator or online calculator by substituting these trigonometric functions. Once you calculate the values, you will have the answers in kilometers.

each boat traveled 51km

[email protected] = 51(-cos26.1,-sin26.1)
[email protected] = 51(-cos38.8,-sin38.8)
That will give you the (x,y) coordinates of each boat

Then subtract as needed to get the desired distances