# A cylinder and a cone have equal radii of their base and equal heights.

Question:

A cylinder and a cone have equal radii of their base and equal heights. If their curved surface area are in the ratio 8:5, show that the radius of each is to the height of each as 3:4.

Solution:

It is given that the base radius and the height of the cone and the cylinder are the same.

So let the base radius of each is 'r' and the vertical height of each is 'h'.

Let the slant height of the cone be 'l'.

The curved surface area of the cone = πrl

The curved surface area of the cylinder =

It is said that the ratio of the curved surface areas of the cylinder to that of the cone is 8:5

So,

$\frac{2 \pi r h}{\pi r l}=\frac{8}{5}$

$\frac{2 h}{l}=\frac{8}{5}$

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

But we know that

$\mathrm{l}=\sqrt{\mathrm{r}^{2}+\mathrm{h}^{2}}$

$\frac{h^{2}}{\sqrt{r^{2}+h^{2}}}=\frac{4}{5}$

Squaring on both sides we get:

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

$\mathrm{r}^{2} / \mathrm{h}^{2}+1=25 / 16$

$r^{2} / h^{2}=25 / 16-1$

$r^{2} / h^{2}=9 / 16$

r/h = 3/4

Hence it is shown that the ratio of the radius to the height of the cone as well as the cylinder is: 3: 4