#### To determine

**To find:**

Work required to raise the elevator from the basement to the third floor.

#### Answer

Work required to raise the elevator from the basement to the third floor is 1,03,500 ft.lb.

#### Explanation

**1) Concept:**

Work done in moving an object from a to b

**2) Formula:**

Work done in moving an object from a to b

∫abfxdx

Work=Force ×Distance

**3) Given:**

Weight of elevator is 1600 lb.

Length of cable is 200 ft.

Weight of cable is 10 lb/ft.

**4) Calculations:**

Let’s place the origin at the top and the x-axis pointing downward as in the figure.

Divide the cable into small parts with length ∆x.

If xi* is a point in the ith sub interval, then all points in the interval are lifted by approximately the same amount, namely xi*.

Now, the weight of the cable is 10 pounds per foot.

So the weight of the ith part is (10 lb/ft)(∆x ft)=10∆x lb.

Thus, work done on the ith part in foot-pounds is

10∆x·xi*=10xi*∆x

The total work done by adding all these approximations and letting the number of parts become large (so ∆x→0),

W=limn→∞∑i=1n10xi*∆x=∫03010x dx

After integrating,

W=5x2300=5302-0=4500 ft·lb

Now, the work needed to raise the elevator alone is =Force×distance

=1600 lb×30 ft=48000 ft·lb.

Here, the elevator is suspended by a 200 ft cable having weight 10 lb/ft.

Therefore, the work needed to raise the bottom 200-30=170 ft of cable is

=170×10×30=51000 ft·lb.

Therefore, the total work is

W=4500+48000+51000=103500 ft·lb.

**Conclusion:**

Work required to raise the elevator from the basement to the third floor is 1,03,500 ft.lb.