Solve each problem. See Examples 1 and 2.

Flying Distance The be...

Question

Solve each problem. See Examples 1 and 2.

Flying Distance The bearing from A to C is N 64° W. The bearing from A to B is S 82° W. The bearing from B to C is N 26° E. A plane flying at 350 mph takes 1.8 hr to go from A to B. Find the distance from B to C.

Accepted Answer

Hey, everyone in this problem. If it takes three hours for a girl to go from her home to the office at a speed of 30 kilometers per hour, then we're asked to determine the distance between the hospital and the office. The bearing from her home to the hospital is north 54 degrees west. From her home to the office is south, 80 degrees west and from the office to the hospital is north 36 degrees east. We're given four answer choices all in kilometers. Option A 64.74. Option B 52.62 option C 88.66 and option D 66.88. So let's start by drawing out what we're given. Ok? So first we have from her home to the office, ok? So we're gonna go ahead and draw her home, ok? As a point and we know that from her home to the office, the bearing is so 80 degrees west. Yes, imagine drawing our kind of axes around this point of home. We're gonna go south which is gonna be pointing straight down and then we're gonna go 80 degrees towards the west. And so our office is going to be over after the office and we know that the bearing to the hospital is north 36 degrees east. So again, we can imagine drawing our axis north is straight upwards east is to the right, we're gonna start going straight upwards and move 36 degrees to the right, gonna look something like this, that angle is 36 degrees. And then finally, we have the bearing from her home to the hospital north 54 degrees west. So if we connect these lines, we can say that the angle between the vertical and that line connecting the home and the hospital, it is gonna be 54 degrees. All right. So that's everything we have here. Now, we know one more thing that's not labeled in our diagram is at 80 degrees. We need to label that first. And that's kind of everything we were given in this problem. So what we're gonna do is try to figure out the distance between the hospital and the office. We are going to label that distance on our diagram. But right now, we don't really have any other information about our trying. Now, we are given information about the trip from home to the office. We know the speed and we know the time it takes. So let's go ahead and find that distance first. And what we're gonna do when we go from home to office is think about the relationship between the speed, the distance and the time. And we know that the speed V is equal to the distance traveled divided by the time it takes, we know we wanna find the distance in this case. So we can rearrange our equation and say that that distance it's gonna be equal to the speed V multiplied by the time T and we have those values. OK. So the speed we travel 30 kilometers per hour multiplied by the time it takes three hours. And this is gonna give us a distance of 90 kilometers. So now we have one side of our triangle, we wanna find the length deep. We don't have the other side. We need to figure out some of these angles. So let's start with the angles. And if we look at our diagram and we're gonna start with the angle on the right hand side, now we have an angle of 54 degrees off the angle inside of our triangle plus 80 degrees and all three of those form a straight line. So we know that those have to add up to 180 degrees. OK? So if we take 180 degrees, we subtract 80 degrees and we subtract 54 degrees, we are gonna be left with 46 degrees. So that gives us that first angle inside of our triangle on the right hand side. For this top angle, we can imagine making some right angle triangle. So if we made some right angle triangles extending out from that point, we know that the angle opposite side d outside of the triangle, we're looking for, we have a right angle, we have an angle of 36 degrees. So the missing angle must be 54 degrees. And on the other side, similarly, we have an angle of 54 degrees. A right angle triangle. And so this missing angle must be 36 degrees. Now, once again, these two angles, 54 degrees and 36 degrees plus the angle we have inside of our triangle form a straight line. So they have to add up to 180 degrees. So again, we take 180 degrees and we subtract these two angles we already have. And what we find is that this is actually a right angle, this is a 90 degree angle. So we have a right angle triangle. Now that we have two of our angles, we can find a third angle that some of the angles in a triangle is 180. So we'll get 180 minus 90 minus 46 degrees and that final angle is gonna be 44 degrees. OK. So when we have problems like this, we really need to think about all the angles we were given and work with them in order to try to determine the angles inside of the triangle that we're dealing. Now that we have our angles, we have this side, we have a side that we're looking for. We can just use our trigonometric ratios and we can use our trigonometric ratios because we have a right angle triangle, it's 90 degrees. So we can choose either one. We know both of the angles. Um not that are not the 90 degree angle because we know all three angles. The only thing we need to make sure we do is include the hypotenuse because that's the side lengths we know. So we're looking at either side or cosine, we're gonna go ahead and choose cosine. And so recall that cosine of the gonna be equal to the adjacent side divided by the hypotenuse, we want the adjacent side to B side D that we're looking for. And so we're gonna choose angle 44 degrees because side D is adjacent to it. So we get cosine of 44 degrees is equal to D divided by the hypotenuse which is 90 kilometers, you multiply both sides by 90 kilometers. And we're gonna get that side length D is equal to 64.7406 kilometers. And that is that final answer we were looking for. That is the distance between the hospital and the office. If we take a look at our answer choices, only round to four significant digits, we can see that the correct answer is going to be option A 64.74 kilometers. Thanks everyone for watching. I hope this video helped see you in the next one.

Solve each problem. See Examples 1 and 2. Flying Distance The bearing from A to C is N 64° W. The bearing from A to B is S 82° W. The bearing from B to C is N 26° E. A plane flying at 350 mph takes 1.8 hr to go from A to B. Find the distance from B to C.

Key Concepts

Bearings

Trigonometric Functions

Distance and Speed Relationship

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