Starting at point A, a ship sails 18.5 km on a bearing of 189°...

Question

Starting at point A, a ship sails 18.5 km on a bearing of 189°, then turns and sails 47.8 km on a bearing of 317°. Find the distance of the ship from point A.

Accepted Answer

Welcome back. Everyone in this problem. A luxury ship starts from point P and covers a distance of 21.7 kilometers with a bearing of 199 degrees. After reaching point Q, the ship turns back and covers a distance of 65.2 kilometers with a bearing of 331 degrees. Calculate the distance of the ship from the starting point for our answer choices. A says it's 81.3 kilometers. B 53.2 kilometers, C 61.4 kilometers and D 75.3 kilometers. Now, if we're going to find the distance of the ship from the starting point, ok. Then first we have to try and draw what's going on here to get a better idea of how to figure out what that distance is. So let's start by showing our bearing. So first we know that we start at a bearing of P, OK? Or restart at point B and from point P, we cover a distance of 21.7 kilometers with a bearing of 199 degrees. Now remember that A bearing is the measurement of the angle clockwise from the north. So if we set up, OK, if we set up our north and south line and our east and west line here, then a bearing of 199 degrees. Let me get rid of this and put p on the other side of the diagram, a bearing of 199 degrees means we're going clockwise from the north 100 and 99 degrees. So it would probably be somewhere here. Ok. So that's our bearing of 199 degrees. Now, we know that from that bearing next, we go to point Q, OK. Point queue for a distance of uh a distance of 21.7 kilometers. Now, where do we go next? Well, after reaching point Q, the ship turns back and covers a distance of 65.2 kilometers with a bearing of 331 degrees. So again, let's set up our reference or north line and then now we're going to go 331 degrees from the north, which would probably be right here. Ok. So let me draw that properly. I want to try to get it to look like a circle as much as possible. It's not the best, but I guess that will work. So let's say that that's our bearing of 331 degrees. Ok. And no, we go to, we go, we're going to go 65.2 kilometers, ok? So let's go 65.2 kilometers probably here. No, we, we, we, we've got to our, our, our final point and now we want to calculate the distance of the ship from the starting point, ok. So we're trying to figure out the distance of the ship from the starting point and now here's where our ship is, ok? Let's say that our ship is at point R and we want that distance. Let me put it in red here. So we want that distance from point R two point P. So is distance right here. Yeah. So the question is how are we going to figure this distance pr out? Well, here, notice that we formed a triangle, OK? For the distances that we've traveled and for this triangle, if we could figure out the angle that's between QR and PQ, then we would have an angle between two sides. And possibly we could use the law of cosines because remember we can use the law of cosines to find the side that is opposite to an angle if we know the value of that anger and the two sides that, that form the anger. OK. So how could we figure out the anger that's between PQ and QR? Well, let's take a look at the information that we have already, we know a part of the anger. OK. We know that from the line QR to the north line would be the difference between 331 degrees and 360 degrees because 360 degrees is a full revolution, 360 minus 331 is 29. So we know that this angle is 29 degrees a part of the angle. Now, how about the other part of the angle? This anger right here? What do we know? Well, if you notice here, let me try to highlight it in blue. Uh If you notice here, this angle forms an altern or this angle is alternate to our angle right here at the point P. OK. And because they are alternate angles, that means both of the angles are the same. So if we can figure out the angle at P that's formed between or north line here or vertical line and uh PQ, then we should be able to figure out the rest of our missing angle. Now, how can we get that? Well, let's think about it. OK. The, the entire bearing is 199 degrees. We already know that it forms 100 and 80 degrees from the north line to the other part of the vertical line. OK. So here is 180 degrees. That means the rest of it must be 199 minus 180 which is 19 degrees. And if this angle is 19 degrees, then the other angle inside our triangle is 19 degrees. So overall our angle is going to be 29 degrees plus 19 degrees, which is equal to 48 degrees. OK. So that angle is 48 degrees inside or a triangle. Now, we have all the information that we need because we can solve using the law of signs. Now, here as is the regular convention when naming triangles, let's call the side opposite to anger QQ, the side opposite to angle RR, that's the 21.7 the side that is 21.7 kilometers, just gonna put R beside it. And let's call the side that's opposite to angle P common P. So P equals 65.2 kilometers. Then by the law of signs or by the law of cosines rather by the law of course, sciences because we have two sides and an angle between them. Then the side opposite to the angle which is Q squared with that in red Q squared is going to be equal to E squared plus R squared. The sum of the squares of the other two sides minus two multiplied by P multiplied by R multiplied by the cosine of the angle between P and R which is Q. Now we can fool these in to solve for the side pr or that is the side Q because this note tells us then that it's going to be equal to 65.2 kilometers squared plus 21.7 kilometers squared minus two multiplied by 65.2 multiplied by 21.7 multiplied by the cosine of 48 degrees. Because that's the angle between them. That means if Q squared is equal to this expression, Q is going to be equal to the square root of that expression. OK. So let me just rewrite that expression here. And now to figure out the value of Q, we can find the square root of our expression, We could put it exactly into our calculator. OK? And that will help us to get an, a more exact value. Now, when you put this into your calculator to solve for Q, that is the side pr you should find that the side Q is going to be equal to F or well, approximately equal to 53.2 kilometers. And that's because if we recall in our answer choices, all of our answers were rounded to the first decimal place. Therefore, the distance between the ship from the start, the distance of the ship rather from the starting point is 53.2 kilometers. If we look back on our answer choices, that would be answer. Choice B thanks a lot for watching everyone. I hope this video helped.

Starting at point A, a ship sails 18.5 km on a bearing of 189°, then turns and sails 47.8 km on a bearing of 317°. Find the distance of the ship from point A.

Key Concepts

Bearings

Law of Cosines

Trigonometric Functions

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