[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.