Suppose water is leaking from a tank through a circular hole of area  at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to   , where  is an empirical constant. Determine a differential equation for the height h of water at time t for the cubical tank shown in Figure 1.3.12. The radius of the hole is 2 in., and .

When I look at this problem I immediately think back to Physics and Calculus II and remember that there are three specific functions that allow me to compute height at time (t), the velocity of an object at time (t), and the acceleration of an object at time (t). These are all functions of gravity. If we were looking for an answer in physics or calculus the I would use those formulas. However, since we are in Differential Equations we are looking for a function that can give us the answer so lets look past the gravity functions and onto D.E.

Step one – Write the equation to give us the parameters of the hole:

\f$A_h=\pi (r^2)=\pi(\frac{2}{12})^2=\frac{4}{144}\pi\f$

Step two – Find the Velocity function


Step three – Use Velocity function and our original equation while re-writing it into the correct D.E that we need for the equation

So, now we have our D.E for the height (h) of water at time (t) inside the tank. I had some trouble understanding this one which is why I had to come back and edit it but getting past the idea of being afraid of the  \f$\frac{dh}{dt}\f$  was my first step to understanding this problem.