Question

the ratio of the radii of two cylinders is 2:3 and the ratio of their heights is 3:4. The ratio of their volumes will be​

Answer 1

Given :-

• Radii radio of two cylinders = 2:3
• Ratio of the cylinder's height = 3:4

To find :-

The ratio of their volumes

Volume of a cylinder = πr²h

What is ratio?

Ratio is the comparison of any two quantities which are expressed in it's lowest form.

If the ratio of the radii is 2:3, then the actual radii must be a multiple of a number.

Let that number be x.

Hence the ratio of the radii not reduced to the lowest terms will be : 2x:3x

The same is applied with the height. Let the number be y in this case

Hence the ratio of the heights not reduced to the lowest terms will be : 3y:4y

Therefore,

Radius of first cylinder = 2x

Height of first cylinder = 3y

Radius of second cylinder = 3x

Height of second cylinder = 4y

substituting the values,

RATIO OF THE 2 CYLINDER'S VOLUME :-

$\dfrac{\pi \times {2x}^{2} \times 3y }{\pi \times {3x}^{2} \times 4y }$

$= \dfrac{\pi \times 4 {x}^{2} \times 3y }{\pi \times 9 {x}^{2} \times 4y}$

Cancelling all the common terms,

$\dfrac{ \cancel{\pi} \times 4 \cancel{ {x}^{2}} \times 3 \cancel{y} }{ \cancel{\pi} \times 9 \cancel{ {x}^{2}} \times 4 \cancel{y} }$

$= \dfrac{4 \times 3}{9 \times 4}$

Reducing to the lowest terms :-

$\dfrac{ \cancel{4} \times \cancel{3}}{ \cancel{9} \times \cancel{4}}$

$= \dfrac{1}{3}$

Hence the ratio of the volumes of the two cylinders is :-

1 : 3

Some more formulas :-

Total surface area of the cylinder = 2πrh + 2πr² =≥ 2πr(h+r)

Lateral surface area of the cylinder = 2πrh