00:01
All right.
00:02
So here we have a tall building.
00:04
It's 110 feet.
00:05
We have a 40 foot rope.
00:07
So one i drew in red that has a weight per foot of 0 .5 pounds per foot.
00:15
And our goal is to find the work it takes to pull that rope all the way up to the top of the building.
00:22
So all the way up.
00:24
Okay.
00:26
So basically how we're going to solve this, there is a kind of a cool trick with physics, but we're going to go ahead and do this with calculus.
00:37
Basically, we're going to imagine splitting the rope into little pieces of our weight.
00:44
And each of these pieces need to be pulled up.
00:47
But notice this piece only has to go up less than, say, the piece at the bottom.
00:52
And the piece of the top, well, it's almost there.
00:54
So it's going to take less work to take that piece.
00:57
So we're basically going to sum up the work it would take for each piece.
01:01
Okay.
01:02
Okay, so the work then is going to be equal to the integral.
01:06
And what we're going to do is we're going to define y and y going downward.
01:11
So at y is zero, we don't have any work because we're already at the top.
01:16
But when y is 40, we have to bring that piece all the way up 40 feet.
01:20
Okay, so we are going to take a look at each of the pieces from y equals zero to y equals 40.
01:27
And work is defined as force times distance.
01:30
And the force is the weight.
01:34
So we're basically going to add up all of our weights where this w is for weight.
01:41
So and d, let me just kind of emphasize this information.
01:47
I'll put it down here.
01:48
Okay, they gave us the weight as a function of y, and that equals 0 .5 pounds per foot.
01:56
If i cross multiply or think of it as multiply both sides by d .y, then the little weight contribution is equal to 0 .5 times d .y.
02:09
So each of those i need to lift up.
02:12
So that's my term there and i need to lift it up y, right? because if i'm at the top and i'm at zero, i don't have to lift it up at all.
02:19
But if i'm down at 40, then i go up 40.
02:23
So i just left at y.
02:24
Okay, so we need to get this all in terms of y so we can integrate.
02:29
So we saw already it's 0 .5 d .y.
02:33
So basically we get 0 .5 y, d .y.
02:38
And this is what we're going to integrate.
02:40
So i can pull out the 0 .5.
02:42
I'm going to integrate y from 0 to 40.
02:48
The anti -derivative of y is y squared over two and then i plug in my limits so basically i get uh let's see a half over two is a fourth so 0 .25 times 40 squared and that gives us um uh 400 foot pounds for work to lift that 40 foot rope and pull it all the way to the top of the building.
03:23
So that's pretty cool.
03:25
The next thing we're going to do is kind of do a riemann sum analysis...