Question

Answer the following questions:
2. (i) If the height of two right circular cylinders are in the ratio 3 : 4 and perimeters are in the ratio
1:2, then find the ratio of their volumes.​

Answer 1

Right Circular Cylinder 1:

Height = h

Radius = r

Perimeter = 2πrh

Right Circular Cylinder 2:

Height = H

Radius = R

Perimeter = 2πRH

ATQ:

$\sf \Longrightarrow \dfrac{Height \ of \ Cylinder \ 1}{Height \ of \ Cylinder \ 2} = \dfrac{3}{4}$

$\sf \Longrightarrow \dfrac{h}{H} = \dfrac{3}{4}$

Also ATQ:

$\sf \Longrightarrow \dfrac{Perimeter \ of \ Cylinder \ 1}{Perimeter \ of \ Cylinder \ 2} = \dfrac{1}{2}$

$\sf \Longrightarrow \dfrac{2\pi r h}{2\pi RH} = \dfrac{1}{2}$

2π get's cancelled.

$\sf \Longrightarrow \dfrac{r h}{RH} = \dfrac{1}{2}$

$\sf \Longrightarrow \dfrac{r}{R} \times \dfrac{h}{H} = \dfrac{1}{2}$

Substitute h/H = 3/4 above.

$\sf \Longrightarrow \dfrac{r}{R} \times \dfrac{3}{4} = \dfrac{1}{2}$

$\sf \Longrightarrow \dfrac{3r}{4R} = \dfrac{1}{2}$

$\sf \Longrightarrow6r = 4R}$

$\sf \Longrightarrow \dfrac{r}{R} = \dfrac{4}{6} \ \longmapsto \textcircled{\sf \scriptsize 1}$

Now, the ratio of volumes of Cylinder 1 to Cylinder 2 is:

$\sf \Longrightarrow \dfrac{Volume \ of \ Cylinder \ 1}{Volume \ of \ Cylinder \ 2} = \dfrac{\pi r^2h}{\pi R^2 H}$

π get's cancelled.

$\sf \Longrightarrow \dfrac{Volume \ of \ Cylinder \ 1}{Volume \ of \ Cylinder \ 2} = \dfrac{r^2h}{R^2 H}$

$\sf \Longrightarrow \dfrac{Volume \ of \ Cylinder \ 1}{Volume \ of \ Cylinder \ 2} = \dfrac{r^2}{R^2} \times \dfrac{h}{H}$

$\sf \Longrightarrow \dfrac{Volume \ of \ Cylinder \ 1}{Volume \ of \ Cylinder \ 2} = \dfrac{r}{R} \times \dfrac{r}{R} \times \dfrac{h}{H}$

Substitute r/R = 4/6 and h/H = 3/4 above.

$\sf \Longrightarrow \dfrac{Volume \ of \ Cylinder \ 1}{Volume \ of \ Cylinder \ 2} = \dfrac{4}{6} \times \dfrac{4}{6} \times \dfrac{3}{4}$

$\sf \Longrightarrow \dfrac{Volume \ of \ Cylinder \ 1}{Volume \ of \ Cylinder \ 2} = \dfrac{1}{3}$

∴ Volume of Cylinder 1 : Volume of Cylinder 2 = 1 : 3