Question

A right circular cylinder and a right 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.

Answer 1

$\frac{CSAofcylinder}{CSAofcone}=\frac{8}{5}$

$\Rightarrow \frac{2\pi rh}{\pi rl}=\frac{8}{5}\Rightarrow \frac{h}{l}=\frac{4}{5}\Rightarrow 5h=4l\Rightarrow 5h=4\sqrt{h^2+r^2}$

Squaring both sides, we get

${25h}^2=16(h^2+r^2)\Rightarrow {9h}^2={16r}^2\Rightarrow \frac{h^2}{r^2}=\frac{16}{9}$

$\Rightarrow \frac{h}{r}=\sqrt{\frac{16}{9}}=\frac{4}{3}=4:3or\frac{r}{h}=\frac{3}{4}=3:4$