(II) In what direction should the pilot aim the plane in Probl...

Question

(II) In what direction should the pilot aim the plane in Problem 67 so that it will fly due south?

Accepted Answer

Hey, everyone in this problem, a helicopter is traveling westward at an air speed of 125 kilometers per hour. A crosswind from the northeast begins to affect the helicopter with an average velocity of 35 kilometers per hour. And we're asked in what direction should the helicopter's nose be pointed to minimize lateral drift and maintain a westward course. We have four answer choices. Option A 11 degrees west of north, option B 11 degrees north of west, option C 79 degrees south of west or option D 79 degrees west of south. So let's draw out what we have going on here. Ok. So we want to maintain a westward direction. So we wanna be going straight to the left and we're gonna point our helicopters nose in a certain direction for this to happen. Now, the wind is a crosswind from the northeast. Ok. So it's coming from the northeast. If we draw out our directions, we have north pointing upwards east, pointing to the right. And so this wind is coming from the top right, and it's gonna be moving down into the left. And so we can think about the wind coming down into the left. And so we're gonna need to point our helicopter up into the right a little bit in order for these vectors to add to this nice westward component. Ok. So we're adding these vectors tip to tail to get the resulting vector that's pointing west. Now, we know that the wind speed is 35 kilometers per hour and the air speed of the helicopter is 125 kilometers per hour. So we can add those to our diagram. Our helicopter is pointing up into the right a little bit. We have our wind pointing down into the left and we know that this wind makes an angle of 45 degrees with the horizontal because it is from the northeast. And when we say northeast, we mean exactly northeast perfectly between north and east. So we have that 45 degree angle. All right. So what we wanna do is figure out the direction, so we need to figure out this data value between the other two vectors. That's gonna be the direction the helicopters know at this point. So we have a triangle, we have a couple of side links, we have one angle, we're looking for another angle. No, if we think about the laws that we can use to sort this out, you might be tempted to think about coastline law. In this case, the problem is that the angle we're looking for K is across from a known side value. But this other side value we don't know. OK, we need all three of them in order to find an angle. So instead of doing that, we can use the sine law, OK. And that's gonna be the simplest way to solve this problem. We could do some extra calculations, find some of the missing side link and then use cosine law. But we don't need to do that in this case. So we're called sine law. We have that the ratio of the sine of the angles divided by the corresponding sides is equal in a triangle. So sine of A divided by A is gonna be equal to sine of B divided by B is equal to sine of C divided by C. In this case, we are going to use the side length opposite our angle fade up. So we're gonna have s of our angle theta divided by that side length of 35 kilometers per hour and that's going to be equal to and then we're gonna use the side and angle corresponding that are not. So we have signed of 45 degrees divided by 125 kilometers per hour. Now we have a simple equation with one unknown value. That's the we can go ahead and solve. So rearranging we have sin of data is gonna be equal to 35 divided by 125. OK. The units kilometers per hour will divide out multiplied by sign of 45 degrees. If we take the inverse sign on both sides, we're gonna get that the is approximately equal to 11.42 degrees. OK. So we have this data value of 11.42 degrees. Now, we need to interpret this in terms of the directions. So let's take a look at our diagram. We have theta is 11.42 degrees and let me draw our coordinates right on this corner. So it's clear north, east, south and west. OK. Now, if we start at west, our angle moving towards the north 11 degrees approximately. And so what that means is that we can say that this angle is 11 degrees north of west. And so the correct answer here is going to be option. Thanks everyone for watching. I hope this video you in the next one.

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(II) In what direction should the pilot aim the plane in Problem 67 so that it will fly due south?

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