A ship is sailing due north. At a certain point the bearing of...

Question

A ship is sailing due north. At a certain point the bearing of a lighthouse 12.5 km away is N 38.8° E. Later on, the captain notices that the bearing of the lighthouse has become S 44.2° E. How far did the ship travel between the two observations of the lighthouse?

Accepted Answer

Hello, today we are going to be solving the following word problem using methods of trigonometry. We will be drawing a picture to go along with the word problem as we are reading it. So we are told that a jet ski is heading north. Now, based off of this first phrase, we know that we are dealing with directions. So we are dealing with the north, south, east and west directions. Since a jet ski is traveling north, we know that a jet ski is going from point A to some unknown point B in the north direction. Now at one moment, a bearing of a buoy 872 m away is 53.1 degrees northeast. Now northeast is facing the upper right direction. So that means that from point A, we are looking at a buoy that is located at point C. The angle that this boo buoy is being observed at is at 53.1 degrees. We are also told that the buoy is 872 m away from point A. So we know that the distance is 872. Now, after some time the bearing of the buoy was observed at an angle of 37.2 degrees southeast southeast is the direction facing the lower, right. So at point B, we are observing the buoy at an angle of 37.2 degrees, we want to determine the distance that the jet ski traveled between the times that the bearing of the buoy was measured. Or in other words, we want to determine the distance between point A and point B, which we could label as the side length C. So overall, there are four pieces of information that are given to us. Now, using the, in the information we can form a triangle, we are told that the angle A and angle B of this triangle will be 51 point point one degrees and 37.2 degrees. We know that the side length little B is equal to 872 m. And we want to find the side length C. So how can we go about doing this in order to solve this problem? We're going to need to use the law of science. The law of sines states that we have three ratios that are equal to each other. We have sine of angle A divided by si length A is equal to sine of angle B divided by S length B is equal to sine of the angle C divided by S length C. We can take any two of these ratios and set them equal to each other in order to solve for the side length C. So what we can do is we can take sine of the angle C divided by sin length C. And since we have the angle and sine length of B, we can set that equal to the ratio sine of B is divided by B. But we need to first solve for the angle C. Well, in order to do this, we're going to use the triangle property for the angles of a triangle. This states that the sum of the three angles within a triangle is equal to 180 degrees. So we know that angle A is given to us as 53.1 degrees. We know that angle B is given to us as 37.2 degrees. And we need to solve for angle C but the sum of these three angles should equal to 180 degrees. So let's go ahead and solve for angle C 53.1 plus 37.2 will give us an angle of 90.3 degrees plus angle C is equal to 180 degrees. If we subtract 90.3 to both sides of this equation, we will get an angle C is equal to 89.7 degrees. So this is going to be the angle measure for the angle C. Now using the law of signs, we can state that sine of angle B divided by the side length B is equal to sine of angle C divided by the side length C. Let's go ahead and solve for the side length C. In this equation. The first thing we can do is we can multiply both sides of the equation by the value C. This will leave us with C multiplied by the quantity sine of B divided by B is equal to sign of C. And if we multiply both sides of the equation by the reciprocal of sine B divided by B, we can rewrite the equation. As the following. We will have C is equal to be multiplied by sine of C divided by S of B. We have the three pieces of information that we need to solve. For the side length C. We have the angle C which we just solved for it is 89.7 degrees. We have the angle B which was given to us as 37.2 degrees. And we have the angle side length B which is 872 m. So if we were to plug in these values, we will have C is equal to 872 multiplied by sine of 89.7 degrees divided by sine of 37.2 degrees. When plugging in this value into a calculator, we will get an approximate value for the side length C as 1442.3 m. And this is going to be the distance that the jet ski traveled throughout the time period. And with that being said, the answer to this problem is going to be c 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.

A ship is sailing due north. At a certain point the bearing of a lighthouse 12.5 km away is N 38.8° E. Later on, the captain notices that the bearing of the lighthouse has become S 44.2° E. How far did the ship travel between the two observations of the lighthouse?

Key Concepts

Bearings and Directional Angles

Triangle Formation and Law of Cosines

Relative Motion and Distance Calculation

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