Chapter 8.5, Problem 19E

To determine

The way which will require less work.

Answer to Problem 19E

The bottom will require less work.

Explanation of Solution

Given information :

Height of the tank $=16$ ft.

Radius of the base of the tank $=2$ ft

Weight $62.4{\text{ lb/ft}}^3$

Formula used :

  $W={\displaystyle \underset{a}{\overset{b}{\int }}\left(f\times x\right)}dx$

Calculation :

Let, at any instant water level is at $y$ depth from the rim then work done through out the bottom $={\displaystyle \underset{0}{\overset{6}{\int }}62.4\times 4\pi \left(15+6-y\right)}dy$

  $=62.4\times 4\pi \times {\left[21y-\frac{y^2}{2}\right]}_0^6\approx 84687.3\text{ ft}\text{.lb}$

Work done by the pump over the rim

  $={\displaystyle \underset{0}{\overset{6}{\int }}62.4\times 4\pi \left(15+6\right)}dy=62.4\times 4\pi \times {\left[21y\right]}_0^6$

  $\approx 98801.8$ ft.lb

So, work done at bottom is less than that at rim.