Question

A cable that weighs 8 lb/ft is used to lift 900 lb of coal up a mine shaft 450 ft deep. Find the work done. Show how to approximate the required work by a Riemann sum. (Let x be the distance in feet below the top of the shaft. Enter xi* as xi.)

Answer 1

Given :

Work done by a segment dx of a cable :

dW = 8dx x = 8x dx

So, work done by whole cable :

$W=\int\limits^{450}_0 {8x} \, dx\\\\W = 4x^2|_0^{450}\\\\W = 4( 450^2)\\\\W= 810000\ ft- lb$

Now, work done to bring coal up :

$W'=900\times 450\ ft-lb\\\\W'=405000\ ft-lb$

Therefore, total work done is ( 810000 + 405000 ) = 1215000 ft-lb .

Hence, this is the required solution.