Question

11.
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, then what is the ratio of their base radii to their heights?
(1) 3:4
(2) 5:4
(3) 6:7
(4)9:8​

Answer 1

ANSWER:-

Given:

A right circular cylinder & a right circular cone have equal bases & equal heights. If their curved surface are in the ratio 8:5.

To find:

The ratio of the radius of the base to the height.

CASE1:

Right circular Cylinder:

Let r1 = r

h1 = h

So,

We know curved surface area of the cylinder is 2πrh

•s1= 2πr1h1.............(1)

CASE 2:

Right circular cone:

•r2= r

•h2=h

The curved surface area of the circular cone is s2=πr²l.

Where,

$l = \sqrt{ {r2}^{2} + {h2}^{2} } \\ \\ = > \sqrt{ {r}^{2} + {h}^{2} }$

So,

$= > \pi r \sqrt{ {r}^{2} + {h}^{2} }..................(2)$

Now,

Comparing equation (1) & (2), we get;

$= > \frac{s1}{s2} = \frac{2\pi rh}{\pi r \sqrt{ {r}^{2} + {h}^{2} } } \\ \\ = > \frac{8}{5} = \frac{2h}{ \sqrt{ {r}^{2} + {h}^{2} } } \\ \\ [Squaring \: bot h \: sides]\\ \\ = > { (\frac{8}{5} )}^{2} = \frac{4 {h}^{2} }{ {r}^{2} + {h}^{2} } \\ \\ = > \frac{64}{25} = \frac{4 {h}^{2} }{ {r}^{2} + {h}^{2} } \\ [Cross \: multiplication] \\ = > 64 {r}^{2} + 64 {h}^{2} = 100 {h}^{2} \\ \\ = > 64 {r}^{2} = 100 {h}^{2} - 64 {h}^{2} \\ \\ = > 64 {r}^{2} = 36 {h}^{2} \\ \\ = > 4(16 {r}^{2} = 9 {h}^{2} ) \\ \\ = > 16 {r}^{2} = 9 {h}^{2} \\ \\ = > \frac{ {r}^{2} }{ {h}^{2} } = \frac{9}{16} \\ \\ = > \frac{r}{h} = \sqrt{ \frac{9}{16} } \\ \\ = > \frac{r}{h} = \frac{3}{4}$

Therefore,

r:h= 3:4.

Hence,

The ratio of the radius of the base to their height is 3:4.

Option (1)✓

Hope it helps ☺️