A ship leaves port on a bearing of 34.0° and travels 10.4 mi. ...

Question

A ship leaves port on a bearing of 34.0° and travels 10.4 mi. The ship then turns due east and travels 4.6 mi. How far is the ship from port, and what is its bearing from port?

Accepted Answer

Hey, everyone in this problem, we're told that a ship sails 19.6 miles out of port with a bearing of 321 degrees. After a few hours, the ship makes a turn and starts moving to the west for 8.3 miles. And we're asked to find the bearing of the ship and its distance from the port. We're given four answer choices. Option, a 306 degrees, 25.6 miles. Option B 337 degrees. 25.6 miles. Option C 306 degrees and 11 miles and option D 15 degrees and 25.6 miles. So, what we're gonna do is start by drawing out what we have going on. Ok. So we're gonna start at the port and we are going to have our ship sailing at a bearing of 321 degrees. So we're gonna start at the port. We're gonna draw our corning axes and we can imagine 321 degrees starting from the positive Y axis when we're talking about bearings going in the clockwise direction until we get to 300 and 21 degrees. And this is the direction our ship is going to sail out of the port. And we know it's gonna sail in this direction for 19.6 months. Once it's sailed 19.6 miles, it is then gonna head straight west. So that's gonna be straight left for 8.3 miles. And we wanna find the distance from the port and the bearing. So if we think about the distance, we can draw a nice straight line connecting the final position to the port, we have our distance D that we are looking for. And we also need to find the Barrett. OK. And what we can imagine is if we take a look at our diagram, the bearing is going to be from that positive Y axis again until this bottom line where the distance D is, this is gonna be our bearing B. If we take a look at the triangle we made and we think about the angle, that bottom angle in our triangle being theta, then our bearing bee that we're looking for is just going to be equal to this original bearing that takes us to the other side of our triangle 321 degrees minus that angle inside of the triangle. So minus the, OK. So a couple of things, we need to figure out, we need to figure out this distance D and we wanna figure out this angle theta in our triangle in order to find the bearing beam. Now, if we take a look at the triangle we made right now, we have two sidelines. So we need more information than that to find either this distance D or this other angle theta. So let's start by thinking about this outer triangle. If we think about this outer triangle that is made if we extend this westward line to the right, and we join it with this vertical line and that forms a 90 degree triangle, we know the angle between the vertical and that original line going in the counterclockwise direction is going to be 360 degrees. OK. So the degrees to make the full circle minus 321 degrees. OK? Because we're going now just the opposite direction. So when we add the two, we're gonna get that full circle. So this gives us an angle of 39 degrees. This is a triangle that we've produced on the outside. Since we've produced a triangle, we know that these, some of the angles has to be 180 degrees. We know that this angle is 39 degrees. We know that the top right angle is 90 degrees. That means that the remaining angle must be 51 degrees. We're getting there one more step. We can see that this angle of 51 degrees and the angle inside of our triangle form a straight line when they're put together, they form a straight line, they must also add up to 180 degrees. And so this angle inside of our triangle must be 129 degrees. All right. So we had to do some geometry there. We had to kind of extend some of our shapes. But what we have now is we have two sides and the enclosed angle, two sides in an enclosed angle that's gonna allow us to use the cosine law to find this distance that you were looking for. So let's go ahead and calculate that distance and recall that cosine law tells us that the side we're looking for square, it's gonna be equal to the sum of the squares of the other two sides. So we'll just call them A and B. OK minus two, multiplied by side. A multiplied by side B multiplied by cosine of the angle opposite the side we're looking for. OK. So enclosed by those two other sides, so we have D squared is equal to A squared plus B squared minus two A B cosine of D. And you might see that frequently written with CS instead of D's, we've just changed it to D to represent that distance that we have. Let's go ahead and substitute in our values D squared is gonna be equal to 8.3 miles squared plus 19.6 miles squared minus two, multiplied by 8.3 miles multiplied by 19.6 miles multiplied by cosine of 129 degrees. When we work this out on our calculator, on the right hand side, we take the square root, we're gonna get that, this distance D is equal to about 25.6477 miles. Right? So we have now the distance from the port that this ship is, that's one part of the question. We need to find that bearing as well. OK. So we're looking for the bearing B, we already know we need to calculate this data value from our triangle. And now we can do it and we can do it using sign law. They, we call that the law of signs relates opposite angles and sides to each other. The ratio of the opposite angle in the side to the others. In a triangle, we have angle D and corresponding side D, we have the side opposite of the. So we can use those three pieces of information together to find the. So writing this out mathematically, we have that sign of angle A divided by side A is gonna be equal to sine of angle D divided by side D. These ratios are equal in a triangle substituting in our values, we have sine theta divided by 8.3 miles is equal to sign of 129 degrees divided by 25.6477 miles. We can multiply both sides by 8.3 miles, we get that sign of data is equal to 8.3 miles multiplied by s of 129 degrees divided by 25.6477 miles. The unit of miles divides out, we can take the inverse sign on both sides. And we're gonna get that this angle theta inside of our triangle is about 14.57 degrees. All right. So we have this angle theta, we have to be careful. We are looking for the bearing B, we are not looking for this angle theta for our final answer. OK. So if we look at this angle, theta remember the bearing be measured from that positive vertical axis is gonna be the original bearing 321 degrees minus this angle theta. OK. So we can say that this bearing bee it is going to be equal to 321 degrees minus 14.57 degrees, which gives us a bearing value of 306.43 degrees. So now we have both things that we needed. We have the bearing, we have the distance from the port and we can get back to choosing an answer. We take a look at our answer choices and we round what we found, we can see that the correct answer is going to be option A, the bearing is about 306 degrees and the distance is about 25.6 miles. Thanks everyone for watching. I hope this video helped see you in the next one.

A ship leaves port on a bearing of 34.0° and travels 10.4 mi. The ship then turns due east and travels 4.6 mi. How far is the ship from port, and what is its bearing from port?

Key Concepts

Bearing

Trigonometric Functions

Pythagorean Theorem

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