A cylinder and a cone have equal radii of their bases and equal heights. If their curved surface areas are in the ratio $$8:5$$, show that the radius of each is to the height of each as $$3:4$$.
For cylinder, we have
base radius$$=$$r, height$$=$$h
$$\therefore S_1=$$Curved surface$$=2\pi rh$$
For cone, we have
$$l=\sqrt{r^2+h^2}$$ and, $$S_2=\pi rl=\pi r\sqrt{r^2+h^2}$$
We have,
$$\dfrac{S_1}{S_2}=\dfrac{8}{5}$$
$$\Rightarrow \dfrac{2\pi rh}{\pi r\sqrt{r^2+h^2}}=\dfrac{8}{5}$$
$$\Rightarrow \dfrac{2h}{\sqrt{r^2+h^2}}=\dfrac{8}{5}$$
$$\Rightarrow \dfrac{4h^2}{r^2+h^2}=\dfrac{64}{25}$$
$$\Rightarrow 25h^2=16r^2+16h^2\Rightarrow 9h^2=16r^2\Rightarrow 3h=4r$$
$$\Rightarrow \dfrac{r}{h}=\dfrac{3}{4}$$.