The bearing of a lighthouse from a ship was found to be N 37° ...

Question

The bearing of a lighthouse from a ship was found to be N 37° E. After the ship sailed 2.5 mi due south, the new bearing was N 25° E. Find the distance between the ship and the lighthouse at each location.

Accepted Answer

Hello, today we are going to be solving the following word problem using trigonometry. We will also be drawing a picture while reading the problem. So we are told that a car is traveling on a road that stretches along a north south line. This first sentence tells us that we have a car traveling from one point to another. On a vertical line. We will label the starting position as A and the end position as B no. The second sentence says the bearing of a windmill from the car was measured 48 degrees northeast. So we know that we are looking at a windmill which will we represent as point w at the first point and looking at the windmill, the Barry is measured to be 48 degrees northeast. Now, after traveling 5.2 miles towards the north, another bearing was taken as 64 degrees northeast. So this sentence tells us that the car had traveled 5.2 miles from point A to point B. Another bearing angle was taken at point B and this bearing was taken as 64 degrees northeast. We want to calculate the distance between the car and the windmill at the time when the first bearing was measured. Now again, the first bearing was measured at point A, this means that we want to solve for the length from point A to point W which we can represent as side length B. So how can we solve for side length B using the information that is given to us? Well, since we are representing the problem as a triangle, let's go ahead and first solve for all the angles within the triangle. Now angle A is already given to us as 48 degrees. But we are given the very angle of B, we want the angle inside the triangle. In order to do that, we will have to use the angle for a straight line. Recall that a straight line is equal to 180 degrees. So if we were to subtract 64 degrees from 180 degrees, that will give us the length of angle B within the right triangle. So 180 minus 64 will give us an angle of 116 degrees. So we know that angle B within the triangle is equal to 116 degrees. Now we are going to use the angle some property to sulfur angle W the angle sum property states that the sum of all three angles within a triangle must equal to 180 degrees. So we're going to take angle A which is 48 degrees. Sum it with angle B which is 100 and 16 degrees and sum that with angle W and that should equal to 180 degrees. This equation will allow us to solve for angle W 48 degrees plus 116 degrees will give us 164 degrees. So we have 164 plus W is equal to 180. Subtracting 164 to both sides of the equation will leave us with the angle W is equal to 16 degrees. So we know that the angle W is 16 degrees. Now that we have all three angles of the triangle. How can we solve for the side length be, we can solve for side length B by using the law of signs, the law of sine states that we have sine of angle A divided by the sine length A equal to sine of angle B divided by the S length B is equal to sine of angle W divided by the side length of W. We can take any two of the three ratios and set them equal to each other. In order to solve for the side length B. Now we can take the ratio with respect to B and set it equal to the ratio with respect to W. Since we have the sign length W, the angle W we have the angle B but we are going to solve for the S length B. So using the law of signs, we will have sign of the angle B divided by the S length B is equal to sine of the angle W divided by the S length W let's go ahead and solve for the side length B. Before putting in any values, we can go ahead and cross multiply both of the ratios together that will leave us with S length B multiplied by S of angle W is equal to the S length W multiplied by sine of angle B. In order to solve for B, we're going to divide both sides of this equation by sine of angle W that will leave us with S length B is equal to the S length W multiplied by sine of angle B divided by sine of angle W. Now we will go ahead and enter the values for each of the variables. We know that the S length W is equal to 5.25 0.2 miles. We know that angle B is equal to 100 and 16 degrees. So we have 5.2 multiplied by sine of 116. And we will be dividing this by sine of angle W and we sold for angle W to be 16 degrees. At this point. I would recommend the help of a calculator. And when plugging in this value into a calculator, B will approximately equal to 17.0 miles, this is going to be the distance between the car and the windmill from point A to point W and with that being said, the answer to this problem is going to be b 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.

The bearing of a lighthouse from a ship was found to be N 37° E. After the ship sailed 2.5 mi due south, the new bearing was N 25° E. Find the distance between the ship and the lighthouse at each location.

Key Concepts

Bearing

Trigonometric Functions

Law of Cosines

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