Question

Two cylinders of same volume have their heights in the ratio 1 : 3. Find the ratio of their radii. (1) √3 :1 (2) √2 :1 (3) √5:2 (4) 2: √5

Answer 1

Let heights be h1 and h2
Let radii be r1 and r2
Given h1:h2 = 1:3
$\pi \times {r1}^{2} \times h1 = \pi \times {r2}^{2} \times h2 \\ \frac{ {r1}^{2} }{ {r2}^{2} } = \frac{h2}{h1} = \frac{3}{1} \\ therefore \: \: \frac{r1}{r2} = \sqrt{3}$
Thus, r1:r2 = √3:1
Option: (1)

Answer 2

The ratio of radii of the two cylinders is $\frac{\sqrt{3}}{1}$ .

Step-by-step explanation:

Let the height and radius of 1st cylinder is h1 and r1.

Let the height and radius of 2nd cylinder is h2 and r2.

Given:

• Two cylinders of same volume have their heights in the ratio 1 : 3.

$\rightarrow \: \frac{h1}{h2} = \frac{1}{3} \\ \rightarrow \: v1 = v2$

The volume of 1st cylinder is given by

$\rightarrow \: v1 = \pi \: {(r1)}^{2} h1$

The volume of 2nd cylinder is given by

$\rightarrow \: v2 = \pi \: {(r2)}^{2}h$

As we know that two cylinders have same volume,

$\rightarrow \: v1 = v2 \\ \rightarrow \: \pi {(r1)}^{2} h1 = \pi {(r2)}^{2} h2 \\ \rightarrow \: {( \frac{r1}{r2}) }^{2} = \frac{h2}{h1} \\ \rightarrow \: {( \frac{r1}{r2}) }^{2} = 3 \\ \rightarrow \: \frac{r1}{r2} = \frac{\sqrt{3}}{1}$

So, the ratio of radii of the two cylinders is $\frac{\sqrt{3}}{1}$ .