Transcription
Here's another problem that Kortaggio sent me and, like the other two, this is quite interesting. Although, I think you'll start to see a pattern forming in how these are solved. And that's the whole point, not just to show you neat problems, but also to give you an intuition of how to solve them. So what does it say? It says, "Alice is 100 meters from Bill, and Bill is 300 meters away from Chelsea." Maybe this is a bit of a Clintonian reference, but who knows, or maybe they just wanted people with starting names a, b, and c. So let's draw that out. So Alice is 100 meters from Bill-- all right. Alice is 100 meters from Bill-- and actually he drew a little diagram, so we know that also Chelsea is to the right of Bill. So let's see, the diagram he sent me looks like this. So you have Alice. And then 100 meters away you have Bill. And then 300 meters to the right of that you have Chelsea. This distance right here is 100 meters. And then this distance right there is 300 meters.
And the problem tells us they are all facing east and are standing on the same line. Fair enough. We'll say that's east. So they're all facing in that direction, standing in the same line. They all travel to the east at constant speeds. OK, but they don't say that they're all traveling at the same speed. Each individual speed is constant. So this could be velocity of Alice, lowercase a, velocity of Bill, and then this is the velocity of Chelsea. Alright, they all travel east at constant speeds. In 6 minutes, Alice overtakes Bill. So what has to happen? Let me do this in another color and write down what that equation would look like. So I'll do it in green. In 6 minutes, Alice overtakes Bill. This tells the distance that Alice travels in 6 minutes. So what's the distance that Alice travels in 6 minutes? It would be the velocity of Alice times 6. And she overtakes Bill. So that means she goes 100 meters more than Bill went. So that means she went 100 meters plus the distance that Bill went in that 6 minutes. And the distance that Bill went in 6 minutes is velocity of Bill times 6. Right? This tells us in 6 minutes, Alice went 100 meters more than however far Bill went in 6 minutes. That's that equation there. We got it from this information.
And then the next thing this problem tells us, in another 6 minutes Alice overtakes Chelsea. So from the beginning of our time, it essentially takes 12 minutes. Right? 6 minutes to overtake Bill and then another 6 minutes to overtake Chelsea. So 12 minutes to overtake Chelsea. So we could write that as the distance that Alice travels in 12 minutes is equal to-- Well, she has to overtake Chelsea. That means she made up 400 meters of distance, so that she traveled 400 more meters. So the distance that Alice travels is going to be 400 meters more than the distance that Chelsea travels in 12 minutes. And remember, distance is just equal to rate times time. Right? So in 12 minutes this is how far-- oh sorry-- this is how far Chelsea travels, that's a c. And this is how far Alice travels, and she is going to travel 400 meters more. OK, that's what those two equations tell us.
And what is their question? They're asking us, "How many minutes did it take Bill to overtake Chelsea?" How many minutes did it take Bill to overtake Chelsea? All right, so let's say we want to know time in minutes. How long does it take Bill to overtake Chelsea? Let's say we know that it's time, t. So that tells us that the velocity of Bill times this time is going to be equal to-- for him to overtake Chelsea, he has to travel 300 meters more than her in that same amount of time. So he has to go 300 meters more than how far Chelsea travels in that time. So that's what this is, and then we essentially are just trying to solve for this time. We want to be able to solve for this. Let's solve this equation. Remember, all these equations really have the same form. In order for this person to overtake this person in 6 minutes. That means in 6 minutes she would have to travel 100 meters more than him. In 12 minutes she would have to travel 400 meters more than Chelsea. And then in t minutes-- and that's what we're going to have to solve the problem for-- b is going to have to travel 300 meters more than Chelsea right there. We just solve for t, so let's just solve for t.
So what do you get? You get vbt - vct = 300. I just subtracted Chelsea's velocity times time from both sides of the equation. We can factor out the time, so let's do that. So you get (vb - vc) * time-- I'm just factoring out the time --= 300. Or time = 300/(vb - vc). So if we could somehow figure out what this is equal to, then we would have solved our problem. Let's see if we can use the other information. Let's see, if we solve for vb here and solved for vc here, hopefully the va's will cancel out. And I suspect they will, otherwise this problem would not be solvable. So let's do that. Let's rewrite this equation up here. Let's solve for vb, for Bill's velocity. Let's see, if we divide both sides by 6-- well, let's subtract 100 from both sides. So you have Bill's velocity times 6-- that's just that --= 6 * Alice's velocity - 100. All I did is I subtracted 100 from both sides of this equation and I swapped the sides. Divide both sides by 6. You got the velocity of Bill is equal to the velocity of Alice - 100/6. 100/6 or 50/3-- I'll just keep it as a 100/6. Fair enough.
Let's see if we can do the same thing with this equation. If we subtract 400 from both sides, we get 12 * Chelsea's velocity = 12 * Alice's velocity - 400. Divide both sides of this equation by 12. You get Chelsea's velocity = Alice's velocity - 400/12. All right. So let's if we can substitute this back in for vb, and substitute this back in for vc. And hopefully the va's cancel out. It already looks like they should. So our time, the time that it takes for Bill to overtake Chelsea-- remember that's what t was, it's always good to remind yourself what this whole problem was about to begin with --= 300/vb. Well instead of vb let's write this. Va, the velocity of Alice, - 100/6 - vc. So minus this right here. - va - 400/12. This should simplify to t-- I'll just arbitrarily switch colors. t = 300/va - --what's 100/6? 100/6 is the same thing as 50/3. Distribute the minus sign. - va-- we can already see these are going to cancel out --minus times a minus plus 400/12 is the same thing is 200/6 or 100/3. Right? + 100/3. These two cancel out, va - va. So then we're left with t = 300/(100/3 - 50/3). Right? Or I could say - 50/3 + 100/3, but either way, - 50/3 + 100/3 is just 50/3. Did I do that right? If I divide the top and the bottom by 4, I get this. And then sure, this is a 100/3 and than 50/3. And so when you divide by a fraction that's the same thing is multiplying by its inverse. 300 * 3/50 over 1. And then we can cancel out some terms, so we don't have to multiply big numbers. So if we divide the top and the bottom by 50 that becomes 1, this becomes 6, and then we're left with t = 6 * 3/1, which is 18, and all the time units we were working with the whole time were in minutes. And we're done.
That was a little hairier than the other two, but notice we were able to solve it by just algebraically writing down the information that they actually gave us in the problem and just seeing what we have to solve for and then substituting back in and then miraculously things canceled out. And in general, if you're given a nice problem like this you can normally just, you know, march forward knowing that if you do things correctly probably nice things will happen and terms will cancel out. But I thought you would find that interesting and once again, thanks to Kortaggio for that problem.