Differentiation - Connected Rates of Change

An inverted upright conical coffee filter that is initially filled with coffee before it is drained into a glass pot at a constant rate of \(165\) \(\text{cm}^3\) per minute. The cone has a fixed radius \(a\) \(\text{cm}\) and a fixed height \(k\) \(\text{cm}\). Let \(h\) \(\text{cm}\) and \(r\) \(\text{cm}\) denote the depth of coffee and the corresponding radius of the top surface of the coffee in the filter respectively at time \(t\) minutes.

1. Show that the volume of coffee in the filter, \(V\) \(\text{cm}^3\), can be expressed as \(\frac{\pi a^2 h^3}{3k^2}\) at time \(t\) minutes.
2. How fast is the level of coffee in the filter, \(h\), changing at the instant when the volume of coffee in the filter is \(\frac{1}{8}\) of its initial volume? Express your answer in the form \(\frac{p}{a^2}\), where \(p\) is an exact constant to be determined.
3. State one assumption to be made on the coffee filter for your calculation in part (ii) to be valid.