Question

A cable that weighs 8 lb/ft is used to lift 900 lb of coal up a mine shaft 650 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

Answer:

Total Work=2275000 ft-lb

Explanation:

According to Riemann sum approximate for work needed to lift the cable:

$W= \lim_{n \to \infty} \sum_{i=1}^n8x_i\Delta x$

Sine we have to add 650 terms because distance 650 ft, we will us the integration.

$W=\int\limits^a_b {f(x)} \, dx \\W=\int\limits^{650}_0 {8x} \, dx \\W=[4x^2]_0^{650}\\W=4(650)^2-0^2\\W=4*422500 ft-lb\\W=1690000 ft-lb$

Work done on lifting:

$W_1=900*650\\W_1=585000 ft-lb$

Total Work= $W+W_1$

Total Work=1690000+585000

Total Work=2275000 ft-lb