2021 NYJC P2 Q2

Many industries use rectangular tanks to handle their water, wastewater and chemical storage and processing needs. Because of their shape, rectangular tanks can offer tremendous cost savings for shipping compared to cylindrical tanks.

In fabricating one such industrial-strength storing tank, a customer requires water to flow into the rectangular tank with a horizontal based area \(A\), at a constant rate of \(n\) units of volume per unit time. The water should flow out of the tank through a hole in the bottom, at a rate that is proportional to the square root of the depth of water in the tank. It is also required that when the depth of the water in the tank is \(h\), the level of water in the tank remains constant.

1. Obtain a differential equation for the depth \(x\) at time \(t\).[3]
2. It is known that the tank is filled to a depth of \(4h\) initially. By using the substitution \(x=h{{u}^{2}}\), show that \(u\) satisfies the differential equation \(\frac{2Ah}{n}\frac{\mathrm{d}u}{\mathrm{d}t}=-\frac{u-1}{u}\).[2]
3. By solving this differential equation, find, in terms of \(A\), \(h\) and \(n\), the time needed for the depth to reach \(\frac{16}{9}h\). Describe how \(x\) varies with \(t\).[6]