9709 P32 - Mar 2017 - Q7
2295
A water tank has vertical sides and a horizontal rectangular base, as shown in the diagram. The area of the base is \(2\text{ m}^2\). At time \(t = 0\), the tank is empty and water begins to flow into it at a rate of \(1\text{ m}^3\) per hour. At the same time, water begins to flow out from the base at a rate of \(0.2\sqrt{h}\text{ m}^3\) per hour, where \(h\) m is the depth of water in the tank at time \(t\) hours.
(i) Form a differential equation satisfied by \(h\) and \(t\), and show that the time \(T\) hours taken for the depth of water to reach \(4\) m is given by
\[ T = \int_0^4 \frac{10}{5 - \sqrt{h}} \, dh. \]
(ii) Using the substitution \(u = 5 - \sqrt{h}\), find the value of \(T\).
