Question

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 volume​

Answer 1

Step-by-step explanation:

Let the heights of the cylinders be $h_{1}\:\&\:h_{2}$

And, radii of the base be $r_{1}\:\&\:r_{2}$

Now,

$\frac{h_{1}}{h_{2}} = \frac{3}{4} \: \: and \: \: \frac{2\pi r_{1} }{2\pi r_{2}} = \frac{1}{2}$

$\implies \: \frac{h_{1}}{h_{2}} = \frac{3}{4} \: \: and \: \: \frac{r_{1} }{ r_{2}} = \frac{1}{2}$

Now, their volumes be $V_{1}\:\:\&\:\:V_{2}$

So,

$\implies \: \frac{V_{1}}{V_{2}} = \frac{\pi (r_{1})^{2} h_{1}}{\pi (r_{2}) ^{2} h_{2} }$

$\implies \: \frac{V_{1}}{V_{2}} = \frac{(r_{1})^{2} h_{1}}{(r_{2}) ^{2} h_{2} }$

$\implies \: \frac{V_{1}}{V_{2}} = \bigg( \frac{r_{1}}{r_{2} } \bigg)^{2} . \frac{h_{1}}{h_{2}}$

$\implies \: \frac{V_{1}}{V_{2}} = \bigg( \frac{1}{2} \bigg)^{2} . \frac{3}{4}$

$\implies \: \frac{V_{1}}{V_{2}} = \frac{1}{4} . \frac{3}{4}$

$\implies \: \frac{V_{1}}{V_{2}} = \frac{3}{8}$

$\implies\:V_{1}:V_{2}=3:8$