1 00:00:00,012 --> 00:00:08,223 [music] In our calculus class so far, some of the applications have involved sheep 2 00:00:08,223 --> 00:00:14,933 trapped inside fences and ladders leaning up against buildings. 3 00:00:14,934 --> 00:00:18,021 Well, here's a scene that combines all these things. 4 00:00:18,021 --> 00:00:21,191 I've got a sheep, I've got a fence, I've got a ladder. 5 00:00:21,191 --> 00:00:25,271 Here's a really tall barn. Here's the sun shining down on the ground. 6 00:00:25,271 --> 00:00:30,176 Let's suppose that you're the sheep, stuck behind the fence, but eager to get to the 7 00:00:30,176 --> 00:00:32,992 barn. The really good news is that you're not 8 00:00:32,992 --> 00:00:36,174 just a regular sheep, you're some sort of ninja sheep. 9 00:00:36,174 --> 00:00:39,418 And you've got a supply of a bunch of different ladders. 10 00:00:39,418 --> 00:00:43,091 The question is, what's the shortest ladder that you can use? 11 00:00:43,092 --> 00:00:48,350 You want to pick the shortest ladder that you can get away with, which goes over the 12 00:00:48,350 --> 00:00:52,405 fence and touches the barn. I mean, if the ladder is too short, it 13 00:00:52,405 --> 00:00:56,712 won't even go over the fence. But, even this longer ladder, I mean, it 14 00:00:56,712 --> 00:01:01,362 will get you over the fence, but this ladder is not long enough yet to get you 15 00:01:01,362 --> 00:01:05,493 all the way to the barn. At this point, you're probably feeling 16 00:01:05,493 --> 00:01:08,472 really bad. You now recognize that this is an 17 00:01:08,472 --> 00:01:12,724 optimization problem, but you're thinking, oh no, not again. 18 00:01:12,724 --> 00:01:17,395 Not another optimization problem. Well, let me say a little more about 19 00:01:17,395 --> 00:01:21,441 exactly how tall the fence is, and how far it is from the barn. 20 00:01:21,441 --> 00:01:24,487 Let's say that, that fence is 6.4 meters tall. 21 00:01:24,488 --> 00:01:29,569 And let's say that the fence is 2.7 meters from the barn. 22 00:01:29,569 --> 00:01:35,229 Is something sounding familiar here? Yeah, we've seen numbers like 64 and 27 23 00:01:35,229 --> 00:01:39,111 come up before. We saw them in that hallway problem, 24 00:01:39,111 --> 00:01:42,199 right? What did I want to do in the hallway 25 00:01:42,199 --> 00:01:46,070 problem? I wanted to take some stick and figure out 26 00:01:46,070 --> 00:01:51,866 what the, in this case, longest stick was that I could navigate through this 27 00:01:51,866 --> 00:01:55,365 hallway. But as, remember, the issue was finding 28 00:01:55,365 --> 00:02:00,954 the shortest stick which simultaneously touched this bottom wall, this wall here 29 00:02:00,954 --> 00:02:05,351 in the corner. That's really the exact same problem. 30 00:02:05,351 --> 00:02:09,093 We just have to think about this problem in, in different terms. 31 00:02:09,093 --> 00:02:13,038 Instead of thinking of this as a wall, I think of this as the ground. 32 00:02:13,038 --> 00:02:17,837 Instead of thinking of this is a wall, I think of this as the side of the barn. 33 00:02:17,837 --> 00:02:22,003 And I want to pass through this point. I want to touch this corner, which really 34 00:02:22,003 --> 00:02:24,410 just means I want to clear the fence, you know? 35 00:02:24,411 --> 00:02:29,548 And because this problem happened to be solved by a stick of length 125 36 00:02:29,548 --> 00:02:35,485 millimeters, this problem is going to be solved by a ladder of length 12.5 meters. 37 00:02:35,485 --> 00:02:41,119 And here, and this drawing is actually to scale so I can, I can really demonstrate 38 00:02:41,119 --> 00:02:44,552 it, you know? Here, I position the ladder and yeah, 39 00:02:44,552 --> 00:02:49,752 this, this ladder is in fact long enough to touch the ground, the barn, and just 40 00:02:49,752 --> 00:02:53,477 clear the fence. We've done a ton of stuff this week, but 41 00:02:53,477 --> 00:02:56,772 this is perhaps the most important lesson of all. 42 00:02:56,772 --> 00:03:00,868 At it's core, mathematics is not about just solving problems. 43 00:03:00,868 --> 00:03:06,562 What we've done here is realize an analogy between two seemingly different problems, 44 00:03:06,562 --> 00:03:09,740 alright? The problem of a stick turning a corner 45 00:03:09,740 --> 00:03:13,778 and a ladder clearing a fence. More than about solving problems, 46 00:03:13,778 --> 00:03:18,322 mathematics is really about these kinds of analogies, about being able to recognize 47 00:03:18,322 --> 00:03:21,162 that things that look different are really the same. 48 00:03:21,162 --> 00:03:25,520 People have been doing mathematics for thousands of years, and there's been a ton 49 00:03:25,520 --> 00:03:29,958 of new mathematics developed in that time. So, you might think that with all the new 50 00:03:29,958 --> 00:03:34,431 mathematics, that mathematics would be constantly splintering off and fracturing 51 00:03:34,431 --> 00:03:37,941 into different subjects. But that's not what's happened at all. 52 00:03:37,941 --> 00:03:44,634 This desire to analogize, to unify. To see different parts of the elephant as 53 00:03:44,634 --> 00:03:51,316 the same elephant, right? This tendency has really kept mathematics 54 00:03:51,316 --> 00:03:56,307 together. It means that mathematics doesn't appear 55 00:03:56,307 --> 00:04:02,735 as a bunch of separate subjects, it really appears as a unified whole.