Chapter 9, Problem 34P

A water tank shaped as an ellipsoid (a = 1.5 m, b = 4.0 m, c = 3m) has a circular hole at the bottom, 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 = 9.81m/s². The rate at which the height, h, of the water in the tank changes as the water ?ows out through the hole is given by $\frac{dy}{dt}=\frac{\sqrt{2gy}{r}^2}{ac\left[-1+\frac{{\left(h-c\right)}^2}{c^2}\right]}$ where r_h, is the radius of the hole.

Solve the differential equation for y. The initial height of the water is h = 5.9 m. Solve the problem for different times and find an estimate for the time when h = 0.1 m. Make a plot of y as a function of time.