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. (I know that the return time is 7.67 hours)bearings from north!
To find the time it takes for the boat to return to port and its bearing during the return, we need to break down the problem into different segments and calculate the distance and time for each segment of the journey.
Step 1: Calculate the distance traveled in each segment:
Segment 1: The boat travels at 11 miles per hour for 2 hours. Therefore, the distance traveled in this segment is 11 mph * 2 hours = 22 miles.
Segment 2: The boat travels at 8 miles per hour for 4 hours. Therefore, the distance traveled in this segment is 8 mph * 4 hours = 32 miles.
Segment 3: The boat travels at 7 miles per hour for 2 hours. Therefore, the distance traveled in this segment is 7 mph * 2 hours = 14 miles.
Step 2: Calculate the total distance traveled before the boat returns to port:
Total distance traveled = Distance in segment 1 + Distance in segment 2 + Distance in segment 3
Total distance traveled = 22 miles + 32 miles + 14 miles = 68 miles.
Step 3: Calculate the time it takes for the boat to return to port:
Given that the boat travels at 8 miles per hour and the total distance is 68 miles, we can use the formula:
Time = Distance / Speed.
Time = 68 miles / 8 mph = 8.5 hours.
However, this calculation only gives us the time it takes for the boat to travel the remaining distance to reach the port after segment 3. We need to add the time it took for the boat to travel in segments 1, 2, and 3.
Total time = Time in segment 1 + Time in segment 2 + Time in segment 3 + Time for return.
Total time = 2 hours + 4 hours + 2 hours + Time for return.
Given that the total time is 8.5 hours, we can rearrange the equation to find the time for return:
Time for return = Total time - (Time in segment 1 + Time in segment 2 + Time in segment 3).
Time for return = 8.5 hours - (2 hours + 4 hours + 2 hours) = 8.5 hours - 8 hours = 0.5 hours.
So, the time it takes for the boat to return to port is 0.5 hours, which is equivalent to 30 minutes.
Step 4: Calculate the boat's bearing during the return:
Since the boat is heading directly back to port, its bearing will be the opposite of its last heading, which was 280 degrees. To find the opposite bearing, we add or subtract 180 degrees.
The boat's bearing during the return = 280 degrees + 180 degrees = 460 degrees.
However, since bearings are typically measured from north, we need to convert the bearing to be within 0 to 360 degrees range by subtracting 360 if the bearing is greater than 360.
If the boat's bearing during the return is 460 degrees, then we subtract 360 degrees to get:
Corrected bearing = 460 degrees - 360 degrees = 100 degrees.
Therefore, the boat's bearing as it returns to port is 100 degrees (measured clockwise from north).