calculus question

Question

a freight elevator is loaded with freight at ground level and weighs 2,820 pounds.The elevator is to be pulled up by a 40 foot cable that weighs 15 pounds per linear foot using a pulley system.Approximate the work required to lift the elevator to a height of 24 feet.

Answer 1

As the weight is being pulled up, it will be suspended in the air.

Let y be the vertical distance that is between the suspended 2820 pound weight and the ground.

The length of rope between the pulley and the suspended load = 40 - y

The weight of this length of rope = 15 (40-y) = 600 - 15 y

The total weight of this rope and the suspended load = 2820 + 600 - 15 y = 3420 - 15 y

The work done on this combined weight through a distance dy is

dW = (3420 - 15 y) dy

The total work done in raising the load 24 feet is

 

         24

W = ∫  (3420 - 15 y) dy

         0

 

Evaluate this integral.