Question

A solid cylinder has total surface area of 462 sq. Cm. Curved surface area is 1/3rd of its total surface area. The volume of the cylinder is:

Answer 1

$\Large \mathfrak{ \blue{ \underline{ Question :}}} \\ \\ \sf A \: solid \: cylinder \: has \: total \: surface \: area \\ \sf of \: 462 \: sq \: cm. \: Its \: curved \: area \: is \: one \: third \\ \sf of \: its \: total \: surface \: area. \: Find \: the \: volume \\ \sf of \: the \: cylinder. \\ \\ \Large \mathfrak{ \blue{ \underline{ Solution :}}} \\ \\ \textsf{We have Total Surface Area of cylinder} \\ \qquad \sf = 2πr( h + r ) \: or \: 2πrh + 2πr^{2} \\ \sf 2πrh + 2πr ^{2} = 462 cm ^{2} \qquad ...(i) \\ \\ \sf Where \\ \sf Curved \: Surface \: Area = \frac{1}{3} \times Total \: Surface \: Area \\ \sf \Rightarrow 2πrh = \frac{1}{3} × 462 \: cm^{2} \\\Rightarrow \sf 2πrh = 154 \: cm^{2} \\ \\ \sf{Substituting \: value \: of \: 2πrh \: in \: {eq}^{n} \: (i),} \\ \\ \Rightarrow\sf 154 \: cm^{2} + 2πr^{2} = 462 \: cm^{2} \\ \Rightarrow \sf 2πr^{2} = 462 - 154 \: cm^{2} = 308 \: cm^{2} \\ \Rightarrow \sf r^{2} = \dfrac{308}{2π} \\ \Rightarrow \sf r^{2}= 308 × \dfrac12 × \dfrac7{22} \: cm^{2} \\ \Rightarrow \sf r^{2} = 28 × \dfrac12 × \dfrac72 \: cm^{2} \\ \Rightarrow \sf r = \sqrt{49} \: cm^{2} \\ \Rightarrow \sf r = 7 \: cm \\ \\ \sf \underline{Substituting \: value \: of \: r \: in \: 2πrh} \\ \\ \Rightarrow \sf 2πrh = 154 \: cm^{2} \\ \Rightarrow \sf h = \dfrac{154}{2πr} \\ \Rightarrow \sf h = 154 × \dfrac12 × \dfrac7{22} × \dfrac17 \: cm \\ \Rightarrow \sf h = \dfrac72 \: cm \\ \\ \underline{ \sf Volume \: of \: cylinder : } \\ \sf \qquad= πr^{2}h \\ \qquad \sf = \dfrac {22}7 × 7 \times 7 × \dfrac72 \\ \sf \qquad= 539 \: cm {}^{3}$

Answer 2

${\bold{\underline{\underline{Answer:}}}}$

Let the radius and height of the cylinder be r cm and h cm.

Given : Total surface area of the cylinder

= 462 cm²

∴ 2π r (r + h) = 462 cm² ...(1)

Lateral surface area of cylinder = $\large\tt\dfrac{1}{3}$ × Total surface area of cylinder (Given)

∴ 2π rh = $\large\tt\dfrac{1}{3}$ × 462 = 154 cm² ...(2)

Now, From (1) and (2), we have

$\implies\tt{\dfrac{{2\pi\:r}(r+h)}{{2\pi\:r}}}=\dfrac{462}{154}$

$\implies\tt{\dfrac{\cancel{2\pi\:r}(r+h)}{\cancel{2\pi\:r}}}=\tt{\dfrac{\cancel{462}}{\cancel{154}}}$

$\therefore \dfrac{r+h}{h}=3$

$\implies$ 3h = r+h

$\implies$ r = 3h-h = 2h

$\rightarrow$ r = 2h

Now, from (2) we have

π r × r = 154 cm²

$(\because \:r = 2h)$

$\therefore \dfrac{22}{7}r^{2}=154cm^{2}$

$\implies \:r^{2}={\dfrac{\cancel{154}\times7}{\cancel{22}}}cm^{2}=49cm^{2}$

$\implies$ r = 7 cm

When r = 7 cm then,

$h=\dfrac{r}{2}=\dfrac{7}{2}cm$

$\implies$ $\therefore$ Volume of Cylinder

= $\pi \: r^{2}h = \dfrac{22}{7}(7cm^{2}) \times \dfrac{7}{2} \: cm$

$\longrightarrow$ $<strong>539\:cm^{3}</strong>$