Chapter 9, Problem 36P

A water tank shaped as a cone (R = 2 m, H = 4 m) has a circular hole at the bottom (d = 18mm), as shown. According to Torricelli’s law, the speed v of the water that is discharging from the hole is given by:

$v=\sqrt{2gh}$

where h is the height of the water and g= 9.81 m/s². The rate at which the height, h, of the water in the tank changes as the water flows out through the hole is given by:

$\frac{dh}{dt}=\frac{H^2{d}^2}{4R^2}\frac{\sqrt{2gh}}{h^2}$

Solve the differential equation for h. The initial height of the water is h= 3 m. Solve the problem for different times and find an estimate for the time when h = 0.1 m. Make a plot of h as a function of time.