A heavy rope 50 ft long weighs 04 lbft and hangs over the...

A heavy rope, 50 ft long, weighs 0.4 lb/ft and hangs over the edge of a building 120 ft high. Approximate the required work by a Riemann sum, then express the work as an integral and evaluate it. How much work W is done in pulling the rope to the top of the building? How much work W is done in pulling half the rope to the top of the building?

Transcript

0:00 Hi there.

00:01 So for this problem, we have a 50 feet long, a rope, and this weights 0 .4 pounds per feet.

00:12 And then it hams over a building of 120 feet high.

00:18 So what we need to calculate, let's call this part a of this problem, is how much work is done in pulling the rope to the top of the building.

00:26 So to calculate that, what we need to consider in this case is the word done is just the product between the density that we are given it, the density, which is, well, let me put this into integral.

00:55 We know that the word done is the force integrated over, well, in this case, let's.

01:01 Called the height x, then the force is just in this case the product between the density that we are given for this.

01:19 Let's call that row.

01:21 This m times x, then we need to integrate over x.

01:31 So the solution for this integral, of course, remember that we need to integrate this from zero to 50 because that is the length of this rope.

01:41 So that will be the density times x squared divided by two.

01:45 So we need to evaluate this from zero to 50.

01:48 So if we substitute in here the density that we are given for this, which is 0 .4, we evaluate this at these two points.

01:57 So we will have that that is 50 to the square divided by 2 minus 0 to the square divided by 2.

02:04 So from this we obtain a value of 1 ,000...

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