Chapter 8.5, Problem 7E

To determine

To find the work done lifting the sand this far.

Answer to Problem 7E

1944ft-lb work was done lifting the sand this far.

Explanation of Solution

Given information: A bag of sand originally weighing 144lb was lifted at a constant rate. Sand leaked out at a constant rate. The sand was half gone by the time the bag had been lifted 18ft.

Calculation:

Since the weight of the bag is a linear function of its height, we compute the changes,

  $dx=\left(x1-x0\right)=18$ and $dy=\left(y1-y0\right)=-72$

For, $\left(x0,y0\right)=\left(0,144\right)$ and $\left(x1,y1\right)=\left(18,72\right)$ .

Thus the weight as a function of height is,

  $w=144-4x$

To compute the work, we use the integral,

  $\begin{array}{l}W={\displaystyle {\int }_0^{18}\left(144-4x\right)dx}\\ {\left[1144x-2x^2\right]}_0^{18}\\ 1944\end{array}$

Hence, 1944ft-lb work was done lifting the sand this far.