2011 NJC P1 Q4

A solid is made up of a cone and a cylinder with the same radius, \(r\) cm (see diagram below).

It expands in such a way that the height of the cone is always equal to the radius of the cone. Furthermore, the height of the cylinder, \(h\) cm, varies with \(r\), such that \(h=k\sqrt{r}\), where \(k\) is a real constant. The total volume of the solid is denoted by \(V\) \(cm^3\) and the area of the base of the cylinder is denoted by \(A\) \(cm^2\).

Given that when \(r=4\) cm, \(A\) increases at a rate of \(0.008\pi \) \(cm^2\)/min and \(V\) increases at a rate of \(0.026\pi \) \(cm^3\)/min, determine the value of \(k\).[6]