College Algebra: Solving Other Problems

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Here's an application that you may get a kick out of. It actually involves aviation, so flying and so forth. I actually fly. You know if you have a small plane, a small has a little prop in front. You can sort of see it. And when you come in for a landing, you know, in fact, it's actually very interesting because I do fly. The hardest part about flying is the landing. Everything else is a piece of cake. It's so easy to take off. There's no thinking involved. Flying, once you're up in the sky, it's absolutely trivial. There's nothing to it. But landing, let me tell you, when you have a plane coming down and it gets close to the ground which is very hard, that's scary and that's actually very hard. And, in fact, let me tell you why. Because if you come down and if you go too slowly, then your plane actually does what's called stalls. It doesn't mean that the engine breaks, but you lose all the lift from your plane and then you sort of land. So that's not good. On the other hand, if you're coming down too fast, you see, if you're coming down too fast, then you sort of keep going and might get to the end of the runway and then, you know, you're in trouble that way, too. So you actually have to find that right landing speed. Right? The right landing speed. You don't want to be too slow. You don't want it to stall out. And, on the other hand, you don't want to be too fast and just keep going, you know, right into the snack bar, into the airport. So what you have to do is you have to find the right landing speed.
So here's a fake question about that. Let's take a look. So you're in a small plane and you have to determine the appropriate landing speed and just like I said, it requires basically the speed versus the length of the runway. If you know the length of the runway, you can then figure out what the right speed should be. And if s is the initial landing speed in feet per second, let's say, and L is the length of the runway, then for this particular plane and this formula is not true for all planes. In fact, this formula is not true for any plane. This is a fake-o. But anyway, in this case it's L = .1s^2 - 3s + 22. So let me recap what that means. So L is the length of the runway needed. So notice that if you come in very, very, very fast, s is the speed now of your initial landing. If this was a very big number, then, in fact, the amount of runway you'd need would actually be large as well. You need a lot of room if you're going to come in really, really fast. If this is going to be a smaller number, then, in fact, you'd need less runway. On the other hand you just can't come in any old tiny speed because, like I said, you could easily just fall. So you have to be careful. So now the question is suppose we have an 800-foot long runway, which actually is reasonable. The question is what would the appropriate initial landing speed be for that? So what I want to find out is what's the right s to put in here, what's the right initial landing speed so that, in fact, this number would produce 800 since that would be the amount of runway space I have. So what would I do? Well, I'd set up an equation and the equation would sound like this. And look like this if you're going to watch it or, if you're listening to it, 1s^2 - 3s + 22 and that would have to equal 800, 800 feet if that was in units. You know that. Well, now I want to solve this equation. Well, this is a quadratic equation in s and so, in fact, what I would try to do is bring everything over to one side and da, da, da, da, da, da. We can do that. I'll tell you something that I just personally--this .1 I have to admit sort of admit gives me the willies, you know? It's fine if you like .1's. That's fine. But if not, you could think of that as and then you could sort of multiply everything through both sides, this side and that side by 10. That would clear the denominator here and I would just have an s^2. Of course, there's a price you pay for that. I pick up a factor of 10 here, here and here. But actually, I prefer to do that. You don't have to do that if you don't want to. But I'm going to multiply everything through by 10 just clear all that--the .1 thing and then I would see s^2 - 30s + 220. You know, everything got increased by a factor of 10, and then I have that equals 8000. The exact same quadratic but not under the .1 there and I like it better. Maybe you don't and then you shouldn't do that. Anyway. Well, now I've got to bring this 8000 over to this side because, remember, I want a quadratic to solve, so I want to have everything equal to 0. So I bring this over and it becomes a -8000 so I'd see an s^2 - 30s. And then I have a -8000, but then I add a 220 so that's a net loss negative of 7780 and that all equals 0. Well, now we can try to factor that if you dare. I need 2 numbers that multiply to give 7780, but combine to give 30. This seems way to obscene, way to obscenely large compared to these coefficients and so I'm not even going to bother to try to factor this. You can if you're more young and optimistic than I am. But I'm going to immediately jump to the quadratic formula since I think this is what's going to be required here. So the quadratic formula, of course, you can see it there. It's, in this case, s = -B plus or minus the square root of b^2 - 4ac all over 2a. And those roles are clear. The role of a is just this invisible 1. b is going to be -30. Don't forget that negative sign. And c is going to be this -7780. Well, now if we insert, we see that s = -b so that's negative -30. So that's actually a positive 30. Plus or minus the square root of b^2 so I have to have -30^2, minus 4 times ac, which is 1 times 7780 with this negative, sign here. So that's a negative 7780 which when I multiply by this negative makes that a positive. So I have that huge thing, all divided by 2a. So that's just 2. What sign do I want? Well, you'll notice I have 2 signs here. I've got the plus and the minus. Now you can actually compute this on a calculator if you want or you can just think about it for a second. This is going to be basically 30^2 and then I'm adding stuff to it. So the square root of that is actually exceed 30. So this number is some number much bigger than 30. So 30 minus a number bigger than 30 would be negative divided by 2 is negative and that would give me a negative speed. That means I somehow have to run my plane backwards which you can not do by the way. You think maybe I should just run the propeller the wrong way. To run the propeller the wrong way--don't even think of doing that. So that's actually going to be my extraneous root here. I can't have a negative answer. So, therefore, it must be the positive answer. So the answer must be 30 plus this square root all over 2 and you can just take a calculator if you want and beep, beep, beep, beep. It sounds like I'm making a call on a cellular phone, doesn't it? Hello, Mom? It's really a calculator with the same sound effect and it turns out if you work this out and just compute it, you'd get around a 104.47 feet. I'm sorry, feet per second. So, in fact, my units here are feet per second. So this is the speed I should be traveling, 104.47 feet per second in order to make my 800-foot runway. All I did was take this fact, set it equal to 800 and then just solve these in the quadratic formula; a very direct and unfortunately, somewhat faked example using the quadratic formula to solve a quadratic word problem.
Try these on your own and see what you think. Oh, wait a minute. Don't go away. Wait! I hope you didn't go away. Because I want to tell you there's a bonus in this one. After all that work, if there's no bonus, my god! You wouldn't believe how sad I'd be if I forgot the bonus. Wait. Don't go a way. There's a bonus question. Let's put it up here right now. Can we put that up there right now and let's see? I'm waiting. There we go. There's the bonus. The bonus is how many peanuts are included in those little packets of peanuts when you fly? I fly a lot and you get this great--you're so hungry. You get on the plane; you're waiting for food. You're supposed to get there an hour before, you know? And you get there and you're waiting and they starve you to death. Sometimes they let you take a little bag as you walk down the runway. There's like nothing in it really. By then they give you the peanuts and you're all excited. So how many peanuts are in there? Do you have your guess? Well, let's see. Of course the challenge is just to open the peanuts because they're really hard to open. In fact, this is why they give them to you because it takes just 2 hours to open them. By the time you open them, you're there. It's like, oh, no time. You start to complain and they go, I'm sorry. You have to take off. Thank you for flying though. Thank you very much for flying with us. Oh, wait. This is a real hard one to open. This is one that is permanently sealed. They use it on the space trips because they don't want oxygen to get in there. Of course, you can't take this on a plane because they consider this a weapon. But let's just do that. And let's see how many are included. Two. That's right. There's 2. But, you know, they look so huge. I mean, compared to--if you look out the window like it's the same size as a truck. Because when you're really high up and there--but they're actually quite large peanuts when you think of it that way. Hmm. Very good. Cashew. Yup. I'm flying first class. All right. Try these and now have some fun. I'll see you soon.
Equations and Inequalities
Word Problems with Quadratics - Applications
Solving Other Problems Page [2 of 2]