Question

A heavy rope, 40 feet long, weighs 0.3 lb/ft and hangs over the edge of a building 110 feet high. Let x be the distance in feet below the top of the building.
Find the work required to pull the entire rope up to the top of the building.

1. Draw a sketch of the situation.
We can look at this problem two different ways. In either case, we will start by thinking of approximating the amount of work done by using Riemann sums. First, let’s imagine “constant force changing distance.”

2. Imagine chopping the rope up into n pieces of length ∆x. How much does each little
piece weigh? (This is the force on that piece of rope. It should be the SAME for each
piece of rope.)
3. How far does the piece of the rope located at xi have to travel to get to the top of the
building? (Notice that this is DIFFERENT for each piece of rope; it depends on the
location of the piece.)
4. How much work is done (approximately) to move one piece of rope to the top of the
building?
5. Find the amount of work required to pull the entire rope to the top of the building,
using an integral.

Now, let’s do the same problem, but this time, imagining “constant distance, changing force.” Imagine pulling up the rope a little bit at a time, say, we pull up ∆x feet of rope with each pull.
6. After you have pulled up xi feet of rope, how much of the rope remains to be pulled?
7. What is the force on the remaining amount of rope?
8. Remembering that each pull moves the rope ∆x feet, how much work is done for each
pull?
9. Find the amount of work required to pull the entire rope to the top of the building,
using an integral.

Help 6 to 9 and write it in order with number.

Answer 1