Science:Math Exam Resources/Courses/MATH101/April 2008/Question 02 (d)/Solution 1

A picture is included below.

Let $x_{i}^{*}$ be a sample point. Then the work done on this ith part in foot-pounds is

$W_{i}=2\Delta x\cdot x_{i}^{*}$

where the force is the $2\Delta x$ part and comes from the fact that since the cable weights 2 pounds per foot and the delta x is how much of the cable we are using in the ith part. The $x_{i}^{*}$ part is the distance. Hence, the work done is

$W_{cable}=\lim _{n\to \infty }\sum _{i=1}^{n}2x_{i}^{*}\Delta x=\int _{0}^{500}2x\,dx=\left.x^{2}\right|_{0}^{500}=250000$

However, this does not include the work done to lift the coal itself. This undergoes

$W_{coal}=800*500=400000$

and so the total work done is

$W_{total}=400000+250000=650000$

where this is measured in foot pounds. This completes the proof.