Question

A right circular cylinder ans a cone have equal bases and equal heights. If their curved surface areas are in the ratio $8:5$, show that the ratio between radius of their bases to their height is $3:4$.

Answer 1

Let the radius and height of cylinder and cone be $r$ and $h$

Let slant height of cone be $l$

Curved surface area of cylinder$=2\pi rh$

Curved surface area of cone$=\pi rl$

$Ratio=\frac{CSAofcylinder}{CSAofcone}=\frac{2\pi rh}{\pi rl}=\frac{8}{5}$

$\frac{h}{l}=\frac{4}{5}$

Squaring both side

$\frac{h^2}{l^2}=\frac{16}{25}$

$l^2=\left(\frac{25}{16}\right)h^2$

$h^2+r^2=\left(\frac{25}{16}\right)h^2$

$r^2=\left(\frac{9}{16}\right)h^2$

${\left(\frac{r}{h}\right)}^2={\left(\frac{3}{4}\right)}^2$

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

Hence ratio of radius to height is $3:4$.