Take note that the work done by the force F(x) in moving an object from x=a to x=b is given by:

`W= int_a^b F(x) dx`

To apply this to the problem above, let x be the amount of cable that has been pulled up. So at the basement, the amount of cable that has been pulled up is x=0. And at the third floor , it is x = 30.

And the forces acting in moving object are the weight of the elevator (which is constant) and the weight of the cable at a certain x. To get the weight of the cable a certain x value, subtract x from its total length and multiply that by it weight per foot.

Hence, the total force acting in the moving object is:

`F(x) = 1600 + 10(200-x)`

`F(x)=3600-10x`

Plugging them to the formula of work, we would have:

`W = int_0^30 (3600-10x)dx `

Evaluating the integral, it yields,

`W=(3600x - 5x^2)|_0^30`

`W=(3600*30 - 5*30^2)-(3600*0-5*0^2)`

`W=103500`

Therefore, the work needed to raise the elevator from the basement to the third floor is 103,500 ft-lb.