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 of each is to the height of each as 3:4.

Answer 1

Let radius of cylinder=r

and radius of cone=r

and let height of cylinder=h

and height of cone =h

Now, slant height of cone (l)=$\sqrt{r^2+h^2}$

And curved surface area of cylinder=$2\pi rh$

and of cone=$\pi rl=\pi r\sqrt{r^2+h^2}$

Ratio between their curved surface=8:5

$\therefore \frac{2\pi rh}{\pi r\left(\sqrt{r^2+h^2}\right)}=\frac{8}{5}Squaring,weget\frac{4h^2}{r^2+h^2}=\frac{64}{25}\Rightarrow 100h^2=64r^2+64h^2\Rightarrow 100h^2-64h^2=64r^2\Rightarrow 36h^2=64r^2\therefore \frac{r^2}{h^2}=\frac{36}{64}\Rightarrow \frac{r}{h}=\frac{6}{8}\left(Takingsquareroot\right)\Rightarrow \frac{r}{h}=\frac{3}{4}\therefore Ratiobetweenrandh=3:4Henceproved.$