A right circular cylinder and a right circular cone have equal bases and equal heights. If their curved surface areas are in the ratio $\displaystyle 8$ : $\displaystyle 5$, then find the ratio between the radius of their bases to their height.

Let \(\displaystyle r\) and \(\displaystyle h\) be the radius and height of cylinder as well as cone respectively \[\begin{aligned} & \frac{2 \pi r h}{\pi r l}=\frac{8}{5} \\ & \Rightarrow \frac{2 h}{\sqrt{r^{2}+h^{2}}}=\frac{8}{5} \\ & \Rightarrow 100 h^{2}=64 r^{2}+64 h^{2} \\ & \Rightarrow 36 h^{2}=64 r^{2} \\ & \therefore \frac{r}{h}=\frac{3}{4} \end{aligned} \] Hence the required ratio is $\displaystyle 3$: $\displaystyle 4$