Starting at point X, a ship sails 15.5 km on a bearing of 200°...

Question

Starting at point X, a ship sails 15.5 km on a bearing of 200°, then turns and sails 2.4 km on a bearing of 320°. Find the distance of the ship from point X.

Accepted Answer

Hello, today we are going to be solving the following word problem. So we are told that a luxury ship starts from point P and covers a distance of 12.9 kilometers with the bearing angle of 209 degrees. After reaching point Q, the ship turns back and covers a distance of 5.8 kilometers with the bearing angle of 347 degrees. We want to calculate the distance of the ship from the starting point. So there's a few things that we can draw to help us solve this problem. First, the luxury ship is starting at a point P. Now, since the ship has a bearing angle of 209 degrees, we're going to draw an axis and represent that bearing angle. Now, the bearing angle starts at the top of the Y axis and the bury angle has a total of 209 degrees. The bearing angle tells us the direction that the ship is facing. So the ship is traveling a total of 12.9 kilometers to a secondary point point Q. Now, after reaching point Q, the ship turns back and covers a distance of 5.8 kilometers with a bearing angle of 347 degrees. Now, since the direction has a bearing angle, we're going to draw a secondary axis on point Q. Again, the be angle starts on the Y axis and we're going to represent it with a total of 347 degrees. The bearing angle tells us that the direction that the ship is facing will be in this direction. And it's going to hit an ending point which will label as point R. Now, what we want to do is we want to calculate distance between point Q and point R which we can label with this red line. So how can we go about solving for this problem? One thing I forgot to label is that the distance between Q and R is 5.8 kilometers bite. Notice that we just drew a triangle. Now we know one side of this triangle and we know a second side of this triangle. If we can solve for the angle between these two sides, we can use the law of cosines to help us solve for the distance between P and R. So what we'll need to do is we'll need to solve for the angle with respect to this side. But notice that the angle is split between the sum of two different angles we'll need to solve for Burt for both. So let's go ahead and start by solving for the angle on the right hand side. Now notice that point P has a varying angle of 209 degrees. If we were to subtract 180 degrees from 209 that would give us the angle at the very end of the bearing angle, 209 minus 180 will give us a total of 29 degrees. So we know that the angle at the end of the bearing angle is 29 degrees. Now you may be wondering why it just solve for this angle. Well, what we can do is we can use the property of complementary angles and the property of complementary angle states that the angle located at the end of this bearing angle will be equivalent to the angle on the right hand side of the angle we want to solve. So if this angle is 29 degrees, then the corresponding angle to that will also be 29 degrees. Now that we have the right hand side we want to solve for the left hand side. Now, the bearing angle at point Q is 347 degrees. If we subtract 347 from 360 that will leave us with the angle at the end of the bearing angle. So 360 minus 347 will give us a difference of 13 degrees. So we know that the angle at the end of the buried angle at angle Q or point Q is going to be 13 degrees. Now, the sum of these two angles is going to give us 42 degrees. So we know that the angle between point P or the side of P and Q and the side of Q and R is going to be a total of 42 degrees. Now that we have this angle, we can use the law of cosines to solve for the distance between P and R. Let's go ahead and label this side A, this side B and this side C we want to solve for C. So the law of cosine states that C squared is going to equal to A squared plus B squared minus two multiplied by a multiplied by B multiplied by cosine of the angle with respect to C which is going to be 42 degrees. Now, what we want to do is we want to solve for C using the law of cosines. The first thing we can do is we can take the square root of both sides of the equation that will give us C is equal to the square root of A squared plus B squared minus two multiplied by a multiplied by B multiplied by cosine of 42 degrees. Now, what we are going to do is we're going to plug in the links of A and B. So the length of A is 12.9 kilometers and the length of B is 5.8 kilometers. So that's going to give us the square root of 12.9 squared plus 5.8 squared minus two, multiplied by 12.9 multiplied by 5.8 multiplied by cosine of 42 degrees. Now, at this point, I would recommend using a calculator to help you simplify this quantity. After using AC A calculator, you should end up with c equaling to 9.4 kilometers. This is going to be the total distance that the ship had traveled. And with that being said, the answer to this problem is going to be D so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Starting at point X, a ship sails 15.5 km on a bearing of 200°, then turns and sails 2.4 km on a bearing of 320°. Find the distance of the ship from point X.

Key Concepts

Bearings and Direction

Vector Representation of Displacement

Distance Calculation Using the Pythagorean Theorem

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