Applications of Differentiation Q1

Water is flowing into an inverted upright cone with fixed height \(H\)cm and fixed radius \(R\)cm (as shown in the diagram) such that the radius of the surface of the water is increasing at a constant rate of \(k\)cm s\(^{-1}\).

Find, in terms of \(H\) and / or \(R\), the rate of change of the curved surface area of the cone that is in contact with the water when the cone is filled to \(\frac{1}{8}\) of its capacity.

In the diagram, the angle \(\theta \) subtended by a minor circular arc \(AB\) of fixed radius \(5\)cm is increasing at a constant rate of \(\alpha \) rad s\(^{-1}\). Find the rates of change of the area of the minor segment \(ACB\) at the instants when the area of triangle \(OAB\) is \(7.5\) cm\(^{2}\).