**Question:**

A right circular cylinder and aright circular cone have equal bases and equal heights. If their curved surfaces are in the ratio 8 : 5, determine the ratio of the radius of the base to the height of either of them.

**Solution:**

In cylinder

radius = *r *

height = *h*

Surface area $=2 \times r \times h \times \pi$

In cone

radius = *r *

height = *h*

Slant height $=\sqrt{h^{2}+r^{2}}$

Surface area $=\pi \times r \times \sqrt{h^{2}+r^{2}}$

$\frac{\text { As CSA of cylinder }}{\text { CSA of cone }}=\frac{8}{5}$

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

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

$\frac{4 h^{2}}{h^{2}+r^{2}}=\frac{64}{25}$ (squaring both sides)

$100 h^{2}=64\left(h^{2}+r^{2}\right)$

$36 h^{2}=64 r^{2}$

$\frac{r^{2}}{h^{2}}=\frac{36}{64}$

$\frac{r}{h}=\frac{6}{8}$

$=\frac{3}{4}$

$r: h=3: 4$