A ship leaves port with a bearing of S 40 degrees W. After traveling 7 miles the ship turns 90 degrees and travels on a bearing of N 50 degrees W for 11 miles. At that time, what is the bearing of the ship from the port?
A step by step explanation would really help since I have no idea where to start.
A step by step explanation would really help since I have no idea where to start.
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The "bearings" language does not make sense as stated (I am a sailor, so I understand nautical bearings/headings). I am going to assume the following is really what is meant:
1.) The ship's first *heading* (not bearing) is 40 degrees West of due South.
2.) The ship's second heading is 50 degrees West of due North.
3.) The question wants to know what the bearing is from the port to the ship.
With these assumptions under our belt, let's look at the actual problem.
Since on the first leg, the ship was traveling at 40 W of S, in geometric terms this is (270 - 40) = 230 degrees. (This does *not* match compass bearings, but the standard geometric convention of 0 = along the positive X-axis, 90 is along the positive Y-axis.) So at the end of the first leg, the ship was at position
(7 * cos(230), 7 * sin(230))
On the second leg, at N 50 degrees W, this translates to (90 + 50) = 140 degrees geometrically, the ship traveled *relative to the end of the first leg* to
(11 * cos(140), 11 * sin(140))
so its position at the end of the second leg was
(7* cos(230) + 11 * cos(140), 7 * sin(230) + 11 * sin(140)) = (-12.926, 1.708)
miles relative to the port. If we want the angle, we can take the ratio of these numbers to be the tangent of the angle, so we get:
a = arctan(1.708 / -12.926) = -7.53 degrees
But we also know that tan(x) = tan (x + 180), and we want the bearing with positive Y and negative X, so this is
-7.53 + 180 = 172.47 degrees
Since this is the *geometric* angle, we now have to translate back to a compass bearing, and it is 82.47 degrees West of North , or, if you like, 277.53 degrees on the compass.
1.) The ship's first *heading* (not bearing) is 40 degrees West of due South.
2.) The ship's second heading is 50 degrees West of due North.
3.) The question wants to know what the bearing is from the port to the ship.
With these assumptions under our belt, let's look at the actual problem.
Since on the first leg, the ship was traveling at 40 W of S, in geometric terms this is (270 - 40) = 230 degrees. (This does *not* match compass bearings, but the standard geometric convention of 0 = along the positive X-axis, 90 is along the positive Y-axis.) So at the end of the first leg, the ship was at position
(7 * cos(230), 7 * sin(230))
On the second leg, at N 50 degrees W, this translates to (90 + 50) = 140 degrees geometrically, the ship traveled *relative to the end of the first leg* to
(11 * cos(140), 11 * sin(140))
so its position at the end of the second leg was
(7* cos(230) + 11 * cos(140), 7 * sin(230) + 11 * sin(140)) = (-12.926, 1.708)
miles relative to the port. If we want the angle, we can take the ratio of these numbers to be the tangent of the angle, so we get:
a = arctan(1.708 / -12.926) = -7.53 degrees
But we also know that tan(x) = tan (x + 180), and we want the bearing with positive Y and negative X, so this is
-7.53 + 180 = 172.47 degrees
Since this is the *geometric* angle, we now have to translate back to a compass bearing, and it is 82.47 degrees West of North , or, if you like, 277.53 degrees on the compass.
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First angle is -(90+40) = -130 degrees, second angle is (90+50) = +140 degrees
Final position:
x = 7 cos(-130) + 11 cos(140) = -12.93
y = 7 sin(-130) + 11 sin(140) = 1.71
so the ships distance is sqrt(12.93^2 + 1.71^2) = 13.04 miles
and bearing is arctan( 1.71 / -12.93 ) = 172.47 degrees (W 7.53 N)
Final position:
x = 7 cos(-130) + 11 cos(140) = -12.93
y = 7 sin(-130) + 11 sin(140) = 1.71
so the ships distance is sqrt(12.93^2 + 1.71^2) = 13.04 miles
and bearing is arctan( 1.71 / -12.93 ) = 172.47 degrees (W 7.53 N)