[MUSIC PLAYING]
Welcome to this brainteaser.
So there are two towns.
One we're going to call Mordor.
And one we're going to call Hogsmeade.
We want to build a pier on the shore between the two towns.
But the stingy city council wants
to use as little material as possible.
So how do we minimize the path from Mordor to the pier
and onto Hogsmeade?
So it's going to be somewhere along here.
And you want the shortest path that goes from Hogsmeade
to the pier to Mordor.
Where does--
DR. DEREK MULLER: Hang on.
OK.
So the shortest total path?
DIANNA COWERN: Shortest total path.
JABRIL ASHE: So you're trying to minimize the complete distance.
DIANNA COWERN: Exactly.
You'll want it as close to a straight diagonal
between the two towns as you can make it.
It's funny.
Some people answer questions like this intuitively,
whereas I see the opportunity and set up some equations here.
I would put together an equation
and then minimize the equation.
I just feel like I need to differentiate
this expression here.
I'm stumped.
I feel like just throw it in the center.
DR. DEREK MULLER: Last step of my solution here
is x equals ac on a plus b.
You're welcome.
I'm going to just offset it a bit
to Hogsmeade, because it's further inland.
DIANNA COWERN: You actually got a really nice solution here.
I didn't know that I had the solution.
There's actually a pretty simple solution
to this problem that involves a bit of geometry,
but not much more than that.
I wonder if there's a really nice, beautiful, elegant
solution.
I feel like triangles could be involved here.
Imagine I took Mordor and reflected it
across the shoreline.
I'm going to fold the paper over at the shoreline
and mark it so it goes through the paper,
so that this distance from Mordor to the shoreline,
and then this distance from the shoreline to reflected Mordor
are the same distance.
Now, if we want to find the minimum distance
between Hogsmeade and reflected Mordor, well that's easy.
The shortest path between two points is a straight line.
And now, we have a potential location for a pier.
But will this location minimize the distance from Hogsmeade
to the pier to Mordor?
Well, we can show that the straight line distance is
actually the same distance, because this side
of the triangle is shared, and these two distances
are the same.
Therefore, by similar triangles, this distance
must be the same as this one.
So this location is looking promising.
And now we can see that by any other location of the pier,
like over here, that would make the Hogsmeade-reflected Mordor
distance longer.
And we can show again that it's the same distance
as Hogsmeade-pier-Mordor, which means
that any other location of the pier
would make the path longer.
Therefore, this must be it.
This is the location of the pier.
So that is it.
That's the solution to our problem.
This is where our pier goes.
I love this problem because it's so elegant.
All you need is a little bit of geometry.
Now, you could solve it using more advanced geometry or even
calculus by writing the two distances
as a function of each other and minimizing that.
DR. DEREK MULLER: At a, b, and c,
then the best place to put the pier
is at a location where x is equal to a times c,
which is the distance to there, divided by a plus b.
You're welcome.
But this is just so much more elegant.
So if I've got Mordor here and Hogsmeade
there, what I'm going to do is reflect Hogsmeade.
DR. DEREK MULLER: Oh no.
Reflections, huh?
I should have expected this from a girl who
did a video about mirrors.
It's also solving a problem with a different problem.
We were looking for one distance at the beginning,
but we found a different distance that happened
to be the same distance.
And by similar triangles and using
these really simple arguments, we solved the first problem.
How do you like them apples?
This is getting weird.
But seriously, math is an essential tool for physics.
Algebra brought us constants.
And geometry brought us a way to draw our world with numbers.
And calculus is the only branch of math
that has ever changed the way that I think.
But that's enough mental meandering.
Thank you for watching this video.
I hope you enjoyed it and you subscribe for more brainteasers
or for more physics.
And if you have any interesting math or physics or science
brainteasers, leave them in the comments.
And happy physicsing.
---from the shoreline--
DIANNA COWERN: So you solved it.
I solved the problem.
DIANNA COWERN: Dude, that's awesome.
I can't believe this actually worked.
To be honest though, I had Dianna's assistance
when I was doing some differentiation,
because the chain rule is a long time in my past.
But it came back to me tonight.
DIANNA COWERN: I may not be able to resist
putting that in the video.
That is OK.
This was a team effort.
I mean, can I use props for this easier way?
DIANNA COWERN: Yeah.
Props?
Sure.
So I would take a string, and I would put a pin here,
put the string there.
And then I'd wrap the string around this,
so that I could pull on this side.
I would anchor the string at different points
here and see where I get the most coming out that side.
If I could use props, i would do it like that.