Question

The ratios of the respective heights
and the respective radii of two
cylinders are 1:2 and 2:1 respectively.
Then their respective volumes are in
the ratio.​

Answer 1

$\underline{\underline{\sf SOLUTION : }}$

• First of all assume that radius of first cylinder as r and radius of second cylinder as r'. Similarly, assume that height of first cylinder as h and height of second cylinder as h'.
• In the question we are given that, the ratios of the respective heights and radii of two cylinders are 1:2 and 2:1 respectively. Mathematically ;

$\bullet\: \: \sf \dfrac{r}{r'} = \dfrac{1}{2}$

$\bullet\: \: \sf \dfrac{h}{h'} = \dfrac{2}{1}$

• Now, as in the question it is stated that we need to find the ratio of their respective volumes.

$\dag \: \: \underline{\underline{\textsf{According to the Question Now :}}}$

$:\implies\sf \dfrac{Volume \: of \: 1^{st} \: cylinder}{Volume \: of \: 2^{nd} \: cylinder} = \dfrac{\pi r^2h}{\pi r'^2h'}$

Cancelling π from both numerator as well as from denomintaor we get :

$:\implies\sf \dfrac{Volume \: of \: 1^{st} \: cylinder}{Volume \: of \: 2^{nd} \: cylinder} = \dfrac{r^2h}{r'^2h'}$

$:\implies\sf \dfrac{Volume \: of \: 1^{st} \: cylinder}{Volume \: of \: 2^{nd} \: cylinder} = \dfrac{r^2}{r'^2} \times \dfrac{h}{h'}$

$:\implies\sf \dfrac{Volume \: of \: 1^{st} \: cylinder}{Volume \: of \: 2^{nd} \: cylinder} = \bigg(\dfrac{r}{r'}\bigg)^{2} \times \dfrac{h}{h'}$

$:\implies\sf \dfrac{Volume \: of \: 1^{st} \: cylinder}{Volume \: of \: 2^{nd} \: cylinder} = \bigg(\dfrac{1}{2}\bigg)^{2} \times \dfrac{2}{1}$

$:\implies\sf \dfrac{Volume \: of \: 1^{st} \: cylinder}{Volume \: of \: 2^{nd} \: cylinder} = \dfrac{1}{4} \times \dfrac{2}{1}$

$:\implies\sf \dfrac{Volume \: of \: 1^{st} \: cylinder}{Volume \: of \: 2^{nd} \: cylinder} = \dfrac{2}{4}$

$:\implies \underline{ \boxed{\sf \dfrac{Volume \: of \: 1^{st} \: cylinder}{Volume \: of \: 2^{nd} \: cylinder} = \dfrac{1}{2} }}$

Answer 2

Answer:

Consider the given ratios.

h1:h2=1:2

r1:r2=2:1

We know that the volume of the cylinder

V=πr2h

So, the ratio of volumes of two cylinders

V2V1=πr22h2πr12h1

V2V1=(r2r1)2×h2h1

V2V1=(12)2×21

V2V1=(14)×21

V2V1=12

V1:V2=2:1

Hope it will help you..