91Ó°ÊÓ

Find the work required to pump all of the water out of the top of a cylindrical hot tub that is 8 feet in diameter and 3 feet deep.

Use the fact that the radius of a circle is half of its diameter to obtain the radius of the cylinder. The radius of the cylinder is $\frac{1}{2}·8ft=4ft$.

Consider that the top of the tank is at height $y=3$and the bottom of the tank is at height $y=0$. Draw a diagram that shows a thin representative slice of the tank at some point $y_k^{*}$from the bottom.

The slice at $y_k^{*}$needs to be moved upwards by $d_k=3-y_k^{*}$units in order to be pumped out of the tank.

The slice is a disk with a volume,

$$\begin{gathered} V_k=\mathrm{π}{\left(4\right)}^2∆\mathrm{y} \\ {\mathrm{V}}_{\mathrm{k}}=16\mathrm{π}∆\mathrm{y} \end{gathered}$$cubic feet

and the water density is $\mathrm{Ó\neg }=62.4$pounds per cubic foot.

The work required to lift an object of weight $F$through a distance $d$is given by

$W=Fd=\mathrm{Ó\neg }Vd$.

Therefore, the work required to pump out the representation slice of water is

$W=16\mathrm{π}\left(62.4\right)\left(3-y_k^{*}\right)∆y$.

The integral is defined as,