2019 TJC C2BT2 P2 Q1

A coffee filter is of the shape of a right circular cone. The cone has a base radius \(3\text{ cm}\) and height \(12\text{ cm}\). The coffee powder, as seen in the cross section of the filter below, has a depth of \(4\text{ cm}\). Let \(h\) be the depth of the water in the filter at time \(t\) seconds. Assume that the amount of water in the coffee powder is negligible.

Initially, the filter contains only coffee powder. A machine dispenses hot water into the filter at a constant rate of \(5\text{ cm}^3\) per second and the hot water drips out of the filter at a constant rate of \(2\text{ cm}^3\) per second.

1. Find the exact rate of change of the depth of the water when the depth is \(5\text{ cm}\).[5]
2. Given that \(200\text{ cm}^3\) of hot water will be dispensed into the filter, determine if the hot water will overflow.[2]

[The volume of a cone of base radius \(r\) and height \(h\) is given by \(V = \frac{1}{3}\pi r^2 h\).]