Chapter 28, Problem 28P

A pond drains through a pipe, as shown in Fig. P28.28. Under a number of simplifying assumptions, the following differential equation describes how depth changes with time:

$\frac{dh}{dt}=-\frac{\pi d^2}{4A\left(h\right)}\sqrt{2g\left(h+e\right)}$

where $h=\text{depth}\left(\text{m}\right),t=\text{time}\left(\text{s}\right),d=\text{pipe diameter }\left(\text{m}\right),A\left(h\right)=\text{pond}$ surface area as a function of depth $\left({\text{m}}^2\right),g=g\text{ravitational constant}\left(=9.81{\text{m/s}}^2\right),$ and $e=\text{depth}$ of pipe outlet below the pond bottom (m). Based on the following area-depth table, solve this differential equation to determine how long it takes for the pond to empty given that $h\left(0\right)=6\text{m,}d=0.25\text{m, }e=1\text{m}$

h, m	6	5	4	3	2	1	0
$A\left(h\right),{10}^4{\text{m}}^2$	1.17	0.97	0.67	0.45	0.32	0.18	0