A tank in the form of a right-circular cylinder standing on end is leaking water through a circular hole in its bottom.When friction and contraction of the water at the hole are taken into account,the height h of water in the tank is described by 


where 0 <c <1. How long will it take the tank take to empty if c =0.6? Given that the tank is 10 feet high and has radius 2 feet and the circular hole has radius 0.5 inch and g=32 ft/s^2.

We begin by combining all the constants as follows:

\f[A_{h}=pi*(0.5)^{2}=0.785 ft^{2}\f]

\f[A_{w}=pi*(2)^{2}=12.566 ft^{2}\f]



The DE then becomes;


separating the variables,

\f[ \frac{dh}{\sqrt{h}}=-0.3dt\f]


\f[ \int \frac{dh}{\sqrt{h}}=-0.3\int dt\f]


we can rewrite the above equation as follows

Where L is the constant of integration.

but h(0)=H ; substituting this into the above equation we have

Thus the equation becomes


For the tank to empty , h=0 .therefore,


solving for t ;

\f[t=\frac{\sqrt{H}}{0.15}=\frac{\sqrt{10}}{0.15}=21.08 seconds\f]