A fishing boat leaves port at 4 miles per hour at a bearing of 80∘ for 3 hours, then turns to a bearing of 0∘ at 3 miles per hour for 4 hours, and finally changes to a bearing of 200∘ at 6 miles per hour for 5 hours. At this point, the boat heads directly back to port at a speed of 2 miles per hour. Find the time it takes the boat to return to port as well as the boat's bearing as it does.
Return time:
hours
Return bearing:
A fishing boat leaves port at 4 KNOTS at a HEADING of 80∘ for 3 hours, then turns to a HEADING of 0∘ at 3 KNOTS for 4 hours, and finally changes to a HEADING of 200∘ at 6 KNOTS for 5 hours. At this point, the boat heads directly back to port at a speed of 2 KNOTS. Find the time it takes the boat to return to port as well as the boat's HEADING as it does.
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note 200 deg is 20 degrees west of south. A knot is a nautical mile/hour
North distance = 12 cos 80 - 30 cos 20 + 12 = -14.1
East distance = 12 sin 80 - 30 sin 20 = 1.55
so we turn for home must go 1.55 miles west and 14.1 miles north
let A = angle west of north
tan A = 1.55/14.1 = .11
A = 6.27 degrees west of north
or heading = 360 -6.27 = 353.7 degrees
distance = 14.1/cos 6.27 = 14.2 miles
Oh the time = 14.2/2 = 7.1 hours
A bearing by the way is the direction of something from you. For example that lighthouse bears 10 degrees off our starboard bow or 27 degrees magnetic. The direction you point the fishing boat is heading.
To find the time it takes the boat to return to port, we need to calculate the distance it has traveled in each segment of its journey and then divide it by the speed at which it is returning to port.
In the first segment, the boat travels at 4 miles per hour for 3 hours. The distance traveled is given by:
Distance = Speed * Time = 4 mph * 3 hours = 12 miles.
In the second segment, the boat travels at 3 miles per hour for 4 hours. The distance traveled is:
Distance = Speed * Time = 3 mph * 4 hours = 12 miles.
In the third segment, the boat travels at 6 miles per hour for 5 hours. The distance traveled is:
Distance = Speed * Time = 6 mph * 5 hours = 30 miles.
Now, in the final segment, the boat is traveling at 2 miles per hour and needs to cover the total distance it has traveled so far to return to port. The total distance is:
Total Distance = 12 miles + 12 miles + 30 miles = 54 miles.
To find the time it takes the boat to return to port, we divide the total distance by the speed at which it is returning to port:
Time = Distance / Speed = 54 miles / 2 mph = 27 hours.
Therefore, it takes the boat 27 hours to return to port.
Now let's find the boat's bearing as it returns to port. We need to find the angle between the last segment (bearing 200∘) and the opposite direction (0∘).
Since the bearing changes from 200∘ to 0∘, it means it is making a 180∘ turn to go back to port. Therefore, the boat's bearing as it returns to port is 180∘.
So, the boat's return time is 27 hours, and its bearing as it returns is 180∘.