A cylinder and a cone have equal radii of their bases and equal heights.
Question:

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 and height of each has the ratio 3 : 4.

Solution:

Suppose that the respective radii and height of the cone and the cylinder are r and h.
Then ratio of curved surface areas = 8 : 5
Let the curved surfaces areas be 8x and 5x.

i. e., $2 \pi r h=8 \mathrm{x}$ and $\pi r l=5 \mathrm{x} \Rightarrow \pi r \sqrt{h^{2}+r^{2}}=5 x$

Hence $4 \pi^{2} r^{2} h^{2}=64 x^{2}$ and $\pi^{2} r^{2}\left(h^{2}+r^{2}\right)=25 x^{2}$

$\therefore$ Ratio of curved surface areas $=\frac{4 \pi^{2} r^{2} h^{2}}{\pi^{2} r^{2}\left(h^{2}+r^{2}\right)}=\frac{64}{25}$

$\Rightarrow \frac{4 \pi^{2} r^{2} h^{2}}{\pi^{2} r^{2}\left(h^{2}+r^{2}\right)}=\frac{64}{25}$

$\Rightarrow \frac{4 h^{2}}{\left(h^{2}+r^{2}\right)}=\frac{64}{25}$

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

$\Rightarrow 9 h^{2}=16 r^{2}$

$\Rightarrow \frac{r^{2}}{h^{2}}=\frac{9}{16}$

$\Rightarrow \frac{r}{h}=\frac{3}{4}$

∴ The ratio of the radius and height of the cone is 3 : 4.