1.15: A cone, a sector and rates of change

Figure 4 shows a right circular cone with base radius $\displaystyle r\,\mathrm{cm}$ and slant height $\displaystyle l\,\mathrm{cm}$. It also shows a sector of a circle with radius $\displaystyle R\,\mathrm{cm}$ and arc length $\displaystyle L\,\mathrm{cm}$.

The area of the curved surface of the cone is $\displaystyle A\,\mathrm{cm}^2$.

By considering how the sector can be folded to exactly form the curved surface of the cone, with $\displaystyle R$ and $\displaystyle L$ suitably chosen,

(a) prove that

Sand is poured onto a horizontal surface at a constant rate of $\displaystyle 1.5\,\mathrm{cm}^3\mathrm{s}^{-1}$. The sand forms a pile in the shape of a right circular cone. Its height is always three times the radius of its base.

Given that

where $\displaystyle k$ is a constant,

(b) find the exact value of $\displaystyle k$.

(c) Calculate the rate, in $\displaystyle \mathrm{cm}^2\mathrm{s}^{-1}$ to 3 significant figures, at which the curved surface area of the pile is increasing when the height of the pile is $\displaystyle 24\,\mathrm{cm}$.