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[music] In our calculus class so far, some of the applications have involved sheep

trapped inside fences and ladders leaning up against buildings.

Well, here's a scene that combines all these things.

I've got a sheep, I've got a fence, I've got a ladder.

Here's a really tall barn. Here's the sun shining down on the ground.

Let's suppose that you're the sheep, stuck behind the fence, but eager to get to the

barn. The really good news is that you're not

just a regular sheep, you're some sort of ninja sheep.

And you've got a supply of a bunch of different ladders.

The question is, what's the shortest ladder that you can use?

You want to pick the shortest ladder that you can get away with, which goes over the

fence and touches the barn. I mean, if the ladder is too short, it

won't even go over the fence. But, even this longer ladder, I mean, it

will get you over the fence, but this ladder is not long enough yet to get you

all the way to the barn. At this point, you're probably feeling

really bad. You now recognize that this is an

optimization problem, but you're thinking, oh no, not again.

Not another optimization problem. Well, let me say a little more about

exactly how tall the fence is, and how far it is from the barn.

Let's say that, that fence is 6.4 meters tall.

And let's say that the fence is 2.7 meters from the barn.

Is something sounding familiar here? Yeah, we've seen numbers like 64 and 27

come up before. We saw them in that hallway problem,

right? What did I want to do in the hallway

problem? I wanted to take some stick and figure out

what the, in this case, longest stick was that I could navigate through this

hallway. But as, remember, the issue was finding

the shortest stick which simultaneously touched this bottom wall, this wall here

in the corner. That's really the exact same problem.

We just have to think about this problem in, in different terms.

Instead of thinking of this as a wall, I think of this as the ground.

Instead of thinking of this is a wall, I think of this as the side of the barn.

And I want to pass through this point. I want to touch this corner, which really

just means I want to clear the fence, you know?

And because this problem happened to be solved by a stick of length 125

millimeters, this problem is going to be solved by a ladder of length 12.5 meters.

And here, and this drawing is actually to scale so I can, I can really demonstrate

it, you know? Here, I position the ladder and yeah,

this, this ladder is in fact long enough to touch the ground, the barn, and just

clear the fence. We've done a ton of stuff this week, but

this is perhaps the most important lesson of all.

At it's core, mathematics is not about just solving problems.

What we've done here is realize an analogy between two seemingly different problems,

alright? The problem of a stick turning a corner

and a ladder clearing a fence. More than about solving problems,

mathematics is really about these kinds of analogies, about being able to recognize

that things that look different are really the same.

People have been doing mathematics for thousands of years, and there's been a ton

of new mathematics developed in that time. So, you might think that with all the new

mathematics, that mathematics would be constantly splintering off and fracturing

into different subjects. But that's not what's happened at all.

This desire to analogize, to unify. To see different parts of the elephant as

the same elephant, right? This tendency has really kept mathematics

together. It means that mathematics doesn't appear

as a bunch of separate subjects, it really appears as a unified whole.