Two racing boats set out from the same dock and speed away at the same constant speed of 120 km/h for half an hour (0.500 h), the blue boat headed 23.7° south of west, and the green boat headed 34.4° 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 need to calculate the distances traveled by both boats in the west and south directions.
First, let's find the distance traveled by each boat in the west direction:
The blue boat traveled for 0.500 hours at a constant speed of 120 km/h. Therefore, its distance traveled in the west direction can be calculated using the formula: distance = speed x time.
Distance traveled by the blue boat in the west direction = 120 km/h x 0.500 h = 60 km.
Now, let's find the distance traveled by each boat in the south direction:
To calculate the distance traveled in the south direction, we need to consider the angles at which each boat is pointed. We'll use trigonometry to find the distances.
The blue boat's angle is given as 23.7° south of west. To find the distance traveled in the south direction by the blue boat, we can use the formula: distance = hypotenuse x sin(angle).
Distance traveled by the blue boat in the south direction = hypotenuse x sin(23.7°).
To calculate the hypotenuse, we can use the Pythagorean theorem. The blue boat traveled at a constant speed of 120 km/h for 0.500 hours, so the hypotenuse's length can be calculated using the formula: hypotenuse = speed x time.
Hypotenuse length for the blue boat = 120 km/h x 0.500 h = 60 km.
Now, we can calculate the distance traveled by the blue boat in the south direction:
Distance traveled by the blue boat in the south direction = 60 km x sin(23.7°).
Using a calculator, we find that sin(23.7°) is approximately 0.4040.
Therefore, the distance traveled by the blue boat in the south direction is:
Distance traveled by the blue boat in the south direction ≈ 60 km x 0.4040 ≈ 24.24 km.
Now, let's repeat the same calculations for the green boat.
The green boat traveled for 0.500 hours at a constant speed of 120 km/h, so its distance traveled in the west direction is also 60 km.
The green boat's angle is given as 34.4° south of west. Using the same calculations as before, we find that the distance traveled by the green boat in the south direction is approximately 37.79 km.
To answer the questions:
(a) The blue boat traveled 60 km in the west direction, while the green boat also traveled 60 km in the west direction. Therefore, both boats traveled the same distance in the west direction.
(b) The green boat traveled approximately 37.79 km in the south direction, while the blue boat traveled approximately 24.24 km in the south direction. Therefore, the green boat traveled 37.79 km - 24.24 km = 13.55 km farther south than the blue boat.
To find the distance traveled west by each boat, we need to determine the horizontal component of their velocities.
For the blue boat, the angle south of west is 23.7°. So the horizontal component of its velocity is given by:
cos(23.7°) = westward velocity / 120 km/h
Rearranging the equation, we have:
westward velocity = 120 km/h * cos(23.7°)
Calculating this, we find:
westward velocity = 120 km/h * cos(23.7°) ≈ 108.181 km/h
To find the distance traveled west by the blue boat in the half-hour (0.500 h), we multiply the velocity by the time:
distance west by blue boat = 108.181 km/h * 0.500 h
Calculating this, we find:
distance west by blue boat = 54.091 km
Now let's repeat the same steps for the green boat:
For the green boat, the angle south of west is 34.4°. So the horizontal component of its velocity is given by:
cos(34.4°) = westward velocity / 120 km/h
Rearranging the equation, we have:
westward velocity = 120 km/h * cos(34.4°)
Calculating this, we find:
westward velocity = 120 km/h * cos(34.4°) ≈ 99.682 km/h
To find the distance traveled west by the green boat in the half-hour (0.500 h), we multiply the velocity by the time:
distance west by green boat = 99.682 km/h * 0.500 h
Calculating this, we find:
distance west by green boat = 49.841 km
(a) The blue boat travels 54.091 km west, while the green boat travels 49.841 km west. Therefore, the blue boat travels (54.091 km - 49.841 km) = 4.25 km farther west.
To find the distance traveled south by each boat, we need to determine the vertical component of their velocities.
For the blue boat, the angle south of west is 23.7°. So the vertical component of its velocity is given by:
sin(23.7°) = southward velocity / 120 km/h
Rearranging the equation, we have:
southward velocity = 120 km/h * sin(23.7°)
Calculating this, we find:
southward velocity = 120 km/h * sin(23.7°) ≈ 50.564 km/h
To find the distance traveled south by the blue boat in the half-hour (0.500 h), we multiply the velocity by the time:
distance south by blue boat = 50.564 km/h * 0.500 h
Calculating this, we find:
distance south by blue boat = 25.282 km
Now let's repeat the same steps for the green boat:
For the green boat, the angle south of west is 34.4°. So the vertical component of its velocity is given by:
sin(34.4°) = southward velocity / 120 km/h
Rearranging the equation, we have:
southward velocity = 120 km/h * sin(34.4°)
Calculating this, we find:
southward velocity = 120 km/h * sin(34.4°) ≈ 68.424 km/h
To find the distance traveled south by the green boat in the half-hour (0.500 h), we multiply the velocity by the time:
distance south by green boat = 68.424 km/h * 0.500 h
Calculating this, we find:
distance south by green boat = 34.212 km
(b) The green boat travels 34.212 km south, while the blue boat travels 25.282 km south. Therefore, the green boat travels (34.212 km - 25.282 km) = 8.930 km farther south.