A cylinder and a cone have equal radii of their bases and equal heights. If their curved surface areas are in the ratio $8:5$, the ratio of radius to the height is:

Let the common radius of the base be $r$ and the common height be $h$.
* Curved Surface Area ($\text{CSA}$) of the cylinder = $2\pi rh$
* Curved Surface Area ($\text{CSA}$) of the cone = $\pi rl=\pi r\sqrt{r^2+h^2}$

Given the ratio of their CSAs is $8:5$:

$$\frac{2\pi rh}{\pi r\sqrt{r^2+h^2}}=\frac{8}{5}$$

$$\frac{2h}{\sqrt{r^2+h^2}}=\frac{8}{5}⟹\frac{h}{\sqrt{r^2+h^2}}=\frac{4}{5}$$

Squaring both sides:

$$\frac{h^2}{r^2+h^2}=\frac{16}{25}$$

$$25h^2=16(r^2+h^2)$$

$$25h^2=16r^2+16h^2$$

$$9h^2=16r^2$$

$$\frac{r^2}{h^2}=\frac{9}{16}⟹\frac{r}{h}=\frac{3}{4}$$

Thus, the ratio of the radius to the height is $3:4$.