Question

The ratio of the curved surface areas of two cylinders A and B is 2 3. If the ratio of the volumes of A and B is 4 5. find the ratio of the heights of A and B​

Answer 1

$Let \: r, \: R \: are \: radii \: of \:two\:cylinders\\and \: h ,\:H \:are \: heights .$

$i )ratio \: of \: curved \: surface \:areas = 2:3$

$\frac{C.S.A_{\pink{(first \: cylinder)}}}{C.S.A_{\blue{ (second \: cylider)}}} = \frac{2}{3}$

$\implies \frac{ 2\pi r h }{2\pi RH } = \frac{2}{3}$

$\implies \frac{ r h }{RH } = \frac{2}{3}$

$\implies \frac{ h }{H } = \frac{2R}{3r}\: --(1)$

$ii ) Ratio \: of \: volumes = 4:5$

$\frac{Volume_{\pink{(first \: cylinder)}}}{Volume_{\blue{ (second \: cylider)}}} = \frac{4}{5}$

$\implies \frac{ \pi r^{2} h }{\pi R^{2} H} = \frac{4}{5}$

$\implies \frac{ r^{2} h }{R^{2} H} = \frac{4}{5}$

$\implies \frac{ r^{2}}{R^{2}} \times \frac{2R}{3r} = \frac{4}{5}\: \blue{ [ From \:(1) ]}$

$\implies \frac{2r}{3R} = \frac{4}{5}$

$\implies \frac{r}{R} = \frac{4}{5} \times \frac{3}{2}$

$\implies \frac{r}{R} = \frac{6}{5}$

$\implies \frac{R}{r} = \frac{5}{6} \: --(2)$

$Put\: (2) \: in \: equation (1) , we \:get$

$\implies \frac{ h }{H } = \frac{2R}{3r} \\= \frac{2}{3} \times \frac{5}{6} \\= \frac{5}{9}$

Therefore.,

$\red{ Ratio \:of \: heights } \green { = 5 : 9 }$

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