This next optimization problem that I've been sent seems fun.
I've actually never seen this problem before, and so I'm
even more excited to do it.
So it says, a wire of length 100 centimeters
is cut into two pieces.
Let me draw this wire.
So that's our wire, and it's going to be cut into 2 pieces.
Let me read the whole thing before I draw anything.
One is bent to form a square, and the other is bent to form
an equilateral triangle.
Where should the cut be made if A, the sum of the two areas is
to be a minimum or a maximum?
And it says, allow the possibility of no cut.
Fascinating.
So this is a 100 centimeters long.
Doesn't matter if it's centimeters or meters
or whatever, right?
So it's 100 long.
Could be 100 miles long.
And we're going to cut it at some point.
We'll cut it there.
Well, I shouldn't have done that, because I was
going to use x and y.
But actually, let's use a and b, since I made the
cut look like an x.
So let's say that that is a centimeters from the left
hand side of the wire.
And then what is the distance from the right hand
side of the wire?
I could call it b, but even easier would be to say, if
this is a, then this is going to be 100 minus a.
And they say, I'm going to take one of the wires, let me just
say the left hand wire, and I'm going to take it into, and
I'm going to form a square.
Which is not a rectangle.
A square.
And then the other wire is going to form an
equilateral triangle.
OK.
So let's do it.
And actually, I can already sense that it'll be easier
to manipulate this one.
So I'm going to make the equilateral triangle out of
this side, and make the square out of the side, just because I
have a sense it will make the math a little easier.
So I'm going to take this wire of length a and form an
equilateral triangle out of it, right?
What is the triangle going to look like?
It's going to look like, and I'm going to arbitrarily
switch colors now.
So this is my equilateral triangle.
All the sides are equal.
And how long are each of the sides going to be?
Well, the whole perimeter is a, so each of
these have to be a/3.
And then I also have a square with the leftovers.
I'm not using the the square tool-- so I have it,
I should use it.
Oh, I don't want to do it like that.
OK, so there's my square.
And if the whole perimeter is 100 minus a, what is
the length of each side?
It's going to be this divided by 4, right?
Because all the sides are equal.
So what's this length divided by 4?
100 minus a divided by 4 is 25 minus a/4, right?
So this side is 25 minus a/4, this side is 25 minus a/4, and
this is the same thing, 25 minus a/4, and this
is 25 minus a/4.
And they want us to find what value of a, they say, the
sum of the 2 areas, right?
They want to know what value of a minimizes the sum of the 2
areas and maximize the sum of the 2 areas.
So what are the sum of the 2 areas?
What's the are of this triangle?
I always forget the formula for the area of an equilateral
triangle, so we'll have to kind of derive it right now.
So what I do, is I always draw something like that, and then
we know that this length right here is a/6.
I'm just doing the halfway point.
The whole thing is a/3.
And this is a 30-60-90 triangle, right?
That's 30, this is 60, so this is going to be square
root of 3 times this.
So the height, let me do that here, just so you
know what I'm doing.
This is a some geometry I'm drawing half the triangle.
This would be a/6.
That's a/3.
And so this would be square root of 3 times this.
So square root of 3 over 6 times a.
So what's the area?
Well, the area of this half triangle, is base times
height times 1/2.
So the area of the whole triangle is just going to be
this base times this height.
So the area of the triangle is equal to this base
times that height.
So its area over 6, that's this, times the height, times
square root of 3 over 6, area.
And just so you remember, area of a triangle is
1/2 base times height.
So if we did 1/2 base times height, we would get the area
of this triangle, right?
But the area of the entire triangle is 2
of these triangles.
And so that's why I got rid of the 1/2.
Because it's 2 times 1/2 base times height.
Hopefully that doesn't confuse you.
It's just a little bit of geometry.
Anyway, the area of the triangle is equal to the square
root of 3 over 36 a squared.
And what's the area of the square?
That's easier.
That's just base times height.
So area of the square is equal to 25 minus a over
4, that's just base times height, squared.
And that is equal to 625 minus 2 times-- so it's
20, let's see, 25 over 4 minus 50 over 4 a?
Right.
Minus 50 over 4a, plus a squared over 16.
So that's the area of the square.
But we want to know the combined area of both of them.
So what's the function for the combined area?
So the combined area, and I'll do it in a bold color, area of
both of them, I'll say area combined, that's equal to
the area of the square plus the area of the triangle.
So the area of the square is, let's see.
It's a squared over 16 minus 50 over 4 a plus 625, and we want
to add the area of the triangle, plus square root
of 3 over 36 a squared.
And that's a function of a.
We could it put it like that.
So now we have to optimize it.
And we want to find the minimum point and the maximum
point, if they exist.
Let me clear up some space.
The area of the square is already implicitly there.
I could actually clear up all of this here.
And so what is the derivative?
a prime, the combined-- actually, I hope you
don't mind, I'm going to erase that too.
Because I think I'm going to run out of space and just get
all bunched up, otherwise.
That's not what I wanted to do.
But you know we're talking about.
We already have figured out the area the combined entities.
So now it's just pure calculus.
All the geometry is done.
So what is the derivative of the combined areas as a
function of a is equal to the derivative of this expression?
So it's 2 times 16, 2/16, so it's a over 8.
So let me write this.
a over 8, let me draw a line here, I don't want
things to get confused.
a over 8 minus 50/4, this derivative 625 is just zero,
plus 2 over square root of 3 over 18 a, right?
And so we want to find the minimum, or the maximum,
values for this.
So let's set this equation as equal to 0.
So we get, let me see what we can do.
We could add the a coefficients.
So we have 1/8 plus square root of 3 over 18 a, right, I just
added that to that, minus 50/4.
And we want to set this derivative equal to zero.
So let me switch colors arbitrarily.
And so we get, let me add these two fractions, so 8 and 18.
What's their common denominator?
8 and 9.
8 times 9, is it 72?
And 1/8 is the same thing as 9/72.
And so 18 goes into 72 4 times, so that's plus 4 square roots
of 3, and all that times a is equal to 50/4.
And now we can multiply both sides times the reciprocal of
this coefficient, and we get-- maybe I made a careless
mistake, but the numbers aren't coming out too clean.
But maybe they're not meant to.
a is equal to 50/4 times 72 over 9 plus
4 square roots of 3.
Let's see if we can cancel anything out.
This 4 becomes, this becomes an 18, and so we get a is equal
to 50 times 18 divided by 9 plus 4 square roots of 3.
Now let's see if that makes any kind of sense.
And I think I'm going to have to get the
calculator for this one.
So what should I do first?
Let me figure out what the denominator's value is.
so 3 square root times 4 equals plus 9 is equal to 15.928.
So that a is equal to, what's 50 times 18,
550, 50, 90, right?
5 times 8 is 40, right?
So that's 900 over 15.928, so let me do that, let
me just invert that.
Times 900.
It equals 56.5.
So a is equal to 56.5.
So at least we got a number that makes sense.
It would have been unfortunate if the point that we have to
cut this wire was some number larger, or God forbid we had
a negative number there.
But we actually got a positive number, and it's a number
between 0 and 100.
So we are probably right.
But the question is, is this a minimum value, or is
this a maximum value?
And to figure that out, well, let's see if we're concave
upwards or downwards at this point.
So let's think about it.
Let's take the second derivative of our
combined area function.
So this was our combined area function.
We had set it equal to 0.
Let's take the second derivative of that.
Well that just equals, the derivative of this equals, 1/8
plus square root of 3 over 18, well, times a, but that
just [? switches ?]
coefficient, and then that goes to 0.
So that equals a prime prime as a function of a.
The combined areas are a function of a.
It's just a constant number, and it is a positive
constant number.
So what does that tell us, given that that is a positive
constant Number That tells us that at any value of a, the
second derivative is positive.
The second derivative is positive tells us that we are
concave upwards, really over this whole combined
area function, we're concave upwards.
And how do we know that?
Well, the [UNINTELLIGIBLE]
is positive, the slope is always increasing, right?
And that tells us that any value where the slope a 0,
which was this one, is actually a minimum point.
So a is equal to 56.5.
So if we cut it at 56.5, we are actually at a minimum point.
So how would we maximize the combined value?
Well, maybe here, we have to just take one of the
boundary conditions.
So maybe we could set a as equal to 0, or we could set a
as equal to 100, and figure out what the values are, and one
of those have to be our maximum value.
So what happens when a is equal to zero?
So this was our function to begin with.
What is our combined area function, if a is 0?
Well, this is 0, this is 0, that is 0, so it equals 625.
And what does that translate into?
That's the case where the entire wire is used to make a
square, and none of the wire is used to make a triangle.
In that situation, our combined area, and all the area coming
from the square, would be 625.
Well, what happens when a is equal to 100?
Well then, if this is equal to 100, then the area
of the square is 0.
So we can actually even ignore these terms.
We actually know that this will evaluate to 0, because we are
committing no wire to the square.
And so what is the area of the triangle?
Well, this term right here was the area of the triangle, so it
equaled the square root of 3 over 36 times 100 squared.
So what is that?
Let's see if we can do that.
So 3 square root divided by 36 is equal to, and then 100
squared is 10,000, right?
So times 110, 1, 2, 3.
481.
It equals 481.
So what do we know?
We know that if we cut it exactly at 56.5, and actually
there's a little bit more precision that I didn't pay
attention to, if we cut it at roughly 56.5, the combined
area, actually we haven't even figured out what that combined
area is, but we know it's a minimum point, because we're
concave upward at that point.
And as a fun exercise for you, you might want to plug it into
your calculator, and figure out what the actual area is for
that point, and realize it is the low point.
So that is the minimum value.
That's how we minimize the area of the square or the triangle.
Now, if we committed all the wire to the square, we get an
area of 625, and all of it comes to the square.
If we cut it here, so all of the wire is used to make the
triangle, we get area of 481.
So the maximum value is when a is equal to 0, and our combined
area is 625, because we're doing a square with the whole
wire, and the minimum value is when we actually cut it
right there at 56.5.
So where we commit to 56.5 to the triangle, and we commit,
you know, the remainder, so what is that, 43.5
to the square.
That's when we minimize it.
That was that frankly a fascinating problem, and I hope
I got the problem right, and even if I made a careless
mistake, I think you understand how to do the problem.
I will see you in the next video.