Question

The surface area of a solid cylinder is 462cmsquare and the curved surface area is one third of the total surface area . Find the volume of the cylinder.
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Answer 1

$\bf \to \: \green{{ \underline{given \div }}}$

$\sf \to \: the \: surface \: area \: of \: a \: solid \: cylinder \: = 462cm {}^{2} \\ \\ \sf \to \: the \: curved \: surface \: area \: is \: one \: - third \: the \: total \: surface \: area$

$\bf \to \: \orange{{ \underline{to \: find \div }}}$

$\sf \to \: \blue{{ \underline{the \: volume \: of \: cylinder}}}$

$\bf \to \: { \underline{solution \div }}$

$\sf \to \: total \: surface \: area \: of \: cylinder \: = 462cm. \\ \\ \sf \to \: curved \: surface \: area \: = \dfrac{1}{3} \times (total \: surface \: area) \\ \\ \sf \to \: total \: surface \: area \: = \: 2\pi \: r \: (h + r) \\ \\ \sf \to \: toal \: surface \: area \: = 2\pi \: rh \: + 2\pi \: {r}^{2} = c.s.a \: + 2\pi \: {r}^{2} \\ \\ \sf \to \: t.s.a \: = \dfrac{t.s.a}{3} + 2\pi \: r {}^{2} \\ \\ \sf \to \: t.s.a \: - \: \dfrac{t.s.a}{3} = 2\pi \: r {}^{2} \\ \\ \sf \to \: \dfrac{2t.s.a}{3} = 2\pi \: r {}^{2} \\ \\ \sf \to \: t.s.a \: = 3\pi \: r {}^{2} \\ \\ \sf \to \: 462 \: = 3 \times \dfrac{22}{7} \times {r}^{2} \\ \\ \sf \to \: r \: = 7cm \\ \\ \sf \to \: c.s.a \: = \dfrac{t.s.a}{3} \\ \\ \sf \to \: 2\pi \: rh \: = \dfrac{462}{3} \\ \\ \sf \to \: 2\pi \: rh \: = 154 \\ \\ \sf \to \: 2 \times \dfrac{22}{7} \times 7 \times h \: = 154 \\ \\ \sf \to \: h \: = \dfrac{7}{2} cm \\ \\ \sf \to \: volume \: of \: cylinder \: = \pi \: {r}^{2} h \\ \\ \sf \to \: \dfrac{22}{7} \times 7 \times 7 \times \dfrac{7}{2} = 539 {cm}^{3} \\ \\ \sf \to \orange{{ \underline{volume \: of \: cylinder \: = 539 {cm}^{3} }}}$

Answer 2

GIVEN:-

Total surface area of cylinder = $462cm^2$

FIND:-

VOLUME OF CYLINDER ?

SOLUTION:-

we know,

$Total \: surface \: area \: of \: given \: cylinder \: is = 462cm^2$

$Curved \: surface \: area = \frac{1}{3}\times total$

$surface \: area = \frac{1}{3} \times 462 = 154 {cm}^{2}$

$now, \: total \: surface \: area - curved \: surface \: area$

$= 2 \pi rh + 2 \pi {r}^{2} - 2 \pi rh \\ = > 462 - 154 = 2 \pi {r}^{2}$

$= > 308 = 2 \times \frac{22}{7} \times {r}^{2}$

$= > {r}^{2} = \frac{308 \times 7}{44} = 49$

$r = 7cm$

$now, \: curved \: surface \: area = {154cm}^{2}$

$= > 2 \pi rh = 154$

$= > 2 \times \frac{22}{7} \times 7 \times h = 154$

$= > h = \frac{154}{44} = 3.5cm$

$Volume \: of \: cylinder = \pi {r}^{2} h$

$= \frac{22}{7} \times {7}^{2} \times 3.5$

$= \frac{22}{7} \times 49 \times 3.5 = 539 {cm}^{3}$

$so, \: volume \: of \: cylinder \: is \boxed{ {539cm}^{3} }$