A fishing boat leaves port at 11 miles per hour at a bearing of 210 degrees for 2 hours, then turns to a bearing of 250 degrees at 8 miles per hour for 4 hours, and finally changes to a bearing of 280 degrees at 7 miles per hour for 2 hours. At this point, the boat heads directly back to port at a speed of 8 miles per hour. Find the time it takes the boat to return to port as well as the boat's bearing as it does.
break each section into x- and y-components. Starting at (0,0),
22 @ 210° moves (-11.00,-19.05)
32 @ 250° moves (-30.07,-10.94)
14 @ 280° moves (-13.79,2.43)
Final location: (-54.86,-27.56)
bearing to port = 63°20'
distance = 47.43
time to port = 5.93 hours
distance = 61.39
time to port = 7.67 hours
To find the time it takes for the boat to return to port, we need to calculate the total distance traveled by the boat. Then, we can divide this distance by the boat's speed to get the time taken.
Let's break down the boat's movements step by step:
1. For the first 2 hours, the boat travels at a speed of 11 miles per hour and a bearing of 210 degrees. We can use the formula: distance = speed × time. Therefore, the distance covered in the first 2 hours is: distance1 = 11 mph × 2 hours = 22 miles.
2. For the next 4 hours, the boat travels at a speed of 8 miles per hour and a bearing of 250 degrees. Using the same formula, we can calculate the distance covered in this period: distance2 = 8 mph × 4 hours = 32 miles.
3. Finally, for the last 2 hours, the boat travels at a speed of 7 miles per hour and a bearing of 280 degrees. Again, let's calculate the distance covered: distance3 = 7 mph × 2 hours = 14 miles.
Now, the boat heads directly back to port at a speed of 8 miles per hour. The distance from the current position of the boat to the port is equal to the sum of the distances covered in the previous steps (since the boat is returning to the origin).
Total distance = distance1 + distance2 + distance3 = 22 miles + 32 miles + 14 miles = 68 miles.
Dividing the total distance by the boat's speed of 8 miles per hour will give us the time taken for the boat to return to port:
Time = total distance / speed = 68 miles / 8 mph = 8.5 hours.
Therefore, it takes the boat 8.5 hours to return to port.
To find the boat's bearing as it returns to port, we need to examine the direction from the final position of the boat back to the port. Since the boat is heading directly back to port, the bearing will be the opposite of the last bearing, which was 280 degrees.
Opposite bearing = 280 degrees ± 180 degrees.
Since bearings are measured clockwise from the north, subtracting 180 degrees gives a clockwise direction:
Opposite bearing = 280 degrees - 180 degrees = 100 degrees.
Therefore, the boat's bearing as it returns to port is 100 degrees.