Solve each problem. See Examples 1 and 2.

Distance between Two S...

Question

Solve each problem. See Examples 1 and 2.

Distance between Two Ships Two ships leave a port at the same time. The first ship sails on a bearing of 52° at 17 knots and the second on a bearing of 322° at 22 knots. How far apart are they after 2.5 hr?

Accepted Answer

Hey, everyone in this room, we have two runners that are gonna start at the same point at the same time. One is gonna be on a bearing of 20 degrees at a speed of 15 kilometers per hour and the other runner is going to be on a bearing of degrees at a speed of 20 kilometers per hour. We're asked to determine the distance between both runners after two hours. We're given four answer choices all in kilometers option A option B 40 option C 50 option D 60. The first thing we're gonna do here is draw a little diagram. So we get a sense of what's going on. So we have our initial point here, ok? And we're just gonna draw, I'm just gonna draw in dotted lines, the kind of coordinate plane. So it's easier to see where these angles are. All right. So our first runner starts on a bearing of 20 degrees, ok. Now recall that when we give bearings, ok. This is not the same as giving angles, bearings are measured from the north direction. So this upward vertical direction in a clockwise manner. Ok. So the distance between the positive y axis in runner one or sorry, the angle between the positive Y axis and runner one is going to be 20 degrees, ok? So this is going to be runner one. Now, runner two is on a bearing of 290 degrees. Now, if we start again on this positive Y axis and we go clock wax, OK. When we get to the positive X axis, we've gone to 90 degrees, going to the negative Y axis another 90 degrees. So that's 180 degrees to the negative X axis. Another 90 degrees brings us to 270 degrees and the bearing is 290 degrees. So we're gonna go 20 degrees more than that. And so running two is gonna be running on this angle. We're 20 degrees upwards from the negative XX. All right. So this is runner two and this is kind of what we have going on. Now, we wanna know the distance between these two runners. So what we want to find is this green distance, OK? Between runner one and runner two, we're gonna call this distance D sort of fine distance. D we first need a few other things. We need to see if we can figure out any angles in our triangle. And we actually need to know the length of these two other lines. So let's start with the angle. Now we have that the angle from the positive Y axis to runner one is degrees. We wanna find the angle of this entire corner. OK. The angle that is made between runner one and runner two. Now, runner two makes an angle of 20 degrees from the negative X axis upwards. That means that the remaining angle between runner two and the Y axis is 70 degrees, right? Because they have to add to 90 degrees, right. All right. So this angle has to be 70 degrees. Now, if we have a 70 degree angle plus a 20 degree angle, that's gonna mean this makes a 90 degree angle. OK. So the angle between runner one and runner two is going to be 90 degrees. That's gonna make this a little bit simpler. OK? We know that this is a 90 degree angle. So to find d we can use Pythagorean theorem again, we just need the length of those two sides. Now, how do we find the lengths? Well, we're given the speed and we're given the time that it takes. So we can use both of those to find the distance each runner travels. So for runner one, runner one is traveling 15 kilometers per hour for two hours. So 15 kilometers per hour multiplied by two hours is gonna leave us 30 kilometers. Similarly, for runner two, they're traveling 20 kilometers per hour again for two hours, which means they're gonna travel a distance of 40 kilometers. OK. All right. So the length of the side that we have in quadrant one is going to be 30 kilometers. The length of the siding quadrant two is going to be 40 kilometers. And now we can find the distance D and again, this is now a right angle triangle, we figured that out. And so we can use Pythagorean theorem. And so we have that D squared, the distance we're looking for squared is going to be equal to the sum of the squares of the other two sides. So 30 kilometers squared plus 40 kilometers square simplifying gives us ad square is equal to 2500 kilometers squared. And we can take the square route to get that. The distance D that we're looking for is going to be kilometers. OK? Taking the positive route there because we are talking about distance. All right. So what we found is that the distance between these two runners after two hours is going to be kilometers, which corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.

Solve each problem. See Examples 1 and 2. Distance between Two Ships Two ships leave a port at the same time. The first ship sails on a bearing of 52° at 17 knots and the second on a bearing of 322° at 22 knots. How far apart are they after 2.5 hr?

Key Concepts

Bearing

Velocity and Distance

Law of Cosines

Watch next