Answer
a: $W=625$ ft-lb
b: $W=\frac{1875}{4}$ ft-lb=$468.75$ ft-lb
Work Step by Step
The rope has a length of $50 ft$, we are given a weight of $0.5 \frac{lb}{ft}$, and the rope is elevated $120 ft$ above a building
We first find the force that the rope exerts, then we can solve for a and b
$F=\frac{lb}{ft}*displacement$
$F=(0.5)*(50-x)$
a:
We are looking for the distance pulled up completely, where we use the length of the rope for the upper limit of the integral.
$ W=\int_{0}^{50}(0.5)(50-x)dx = 625$ ft-lb
b:
We are looking for the distance pulled up halfway, where we take the length of the rope and half it, and use that for the upper limit of the integral.
$\frac{50}{2}=25$
$ W=\int_{0}^{25}(0.5)(50-x)dx = \frac{1875}{4}$ ft-lb = $468.75$ ft-lb
