Question

the total surface area of a solid cylinder is 462m2 and its curved surface area is one third of it total surface area. find the volume of a cylinder

Answer 1

$\sf Given \begin{cases} & \sf{Total\:surface\:area\:of\:cylinder = \bf{462\:m^2}} \\ \\ & \sf{CSA = \dfrac{TSA}{3}} \end{cases}$

To find: The Volume of cylinder?

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$\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::}}$

$:\implies\sf CSA_{\:(cylinder)} = \dfrac{TSA}{3}\\ \\ \\ :\implies\sf CSA_{\:(cylinder)} = \cancel{\dfrac{462}{3}}\\ \\ \\ :\implies\sf CSA_{\:(cylinder)} = \bf{154\:m^2}$

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Now,

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$\dag\;{\underline{\frak{As\;we\;know\;that,}}}$

• Total surface area of cylinder is Sum of it's curved surface area and the area of base and top of the cylinder.

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$:\implies\sf TSA = CSA + 2 \pi r^2\\ \\ \\ :\implies\sf 462 = 154 + 2 \pi r^2\\ \\ \\ :\implies\sf 462 - 154 = 2 \times \dfrac{22}{7} \times r^2\\ \\ \\ :\implies\sf 308 = \dfrac{44}{7} \times r^2\\ \\ \\ :\implies\sf r^2 = 308 \times \dfrac{7}{44}\\ \\ \\ :\implies\sf r^2 = 7 \times 7\\ \\ \\ :\implies\sf \sqrt{r^2} = \sqrt{7^2}\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{r = 7\:m}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Radius\:of\:cylinder\:is\: {\textsf{\textbf{7\:m}}}.}}}$

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Calculating Height of cylinder from CSA,

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$\star\;{\boxed{\sf{\pink{Curved\:surface\:area_{\;(cylinder)} = 2 \pi rh}}}}$

$:\implies\sf 154 = 2 \times \dfrac{22}{7} \times 7 \times h\\ \\ \\ :\implies\sf 154 = \dfrac{44}{\cancel{7}} \times \cancel{7} \times h\\ \\ \\ :\implies\sf h = \cancel{\dfrac{154}{44}} \\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{h = \dfrac{7}{2}\:m}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Height\:of\:cylinder\:is\: {\textsf{\textbf{7/2\:m}}}.}}}$

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Now, Finding Volume of cylinder by Substituting given values in Formula,

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Volume of cylinder is given by,

$\star\;{\boxed{\sf{\pink{Volume_{\;(cylinder)} = \pi r^2h}}}}$

$:\implies\sf Volume_{\;(cylinder)} = \dfrac{\cancel{22}}{7} \times 7 \times 7 \times \dfrac{7}{\cancel{2}}\\ \\ \\ :\implies\sf Volume_{\;(cylinder)} = \dfrac{11}{\cancel{7}} \times \cancel{7} \times 7 \times 7\\ \\ \\ :\implies\sf Volume_{\;(cylinder)} = 11 \times 7 \times 7\\ \\ \\:\implies{\underline{\boxed{\frak{\purple{Volume_{\;(cylinder)} = 539\:m^3}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Volume\:of\:solid\:cylinder\:is\: \bf{539^3}.}}}$

Answer 2

$\; \; \; \;{\large{\bold{\sf{\underbrace{\underline{Let's \; understand \; the \; question}}}}}}$

This question says that the total surface area of a solid cylinder is 462m² and its curved surface area is one third of it total surface area We have to find the volume of a cylinder.

${\large{\bold{\sf{\underline{Given \; that}}}}}$

Total surface area of a solid cylinder = 462m²

It's curved surface area is one third of it total surface area means Curved Surface Area = ${\sf{\dfrac{TSA}{3}}}$

${\large{\bold{\sf{\underline{To \; find}}}}}$

Volume of cylinder.

${\large{\bold{\sf{\underline{Solution}}}}}$

Volume of cylinder = 539m³

${\large{\bold{\sf{\underline{Full \; Solution}}}}}$

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~ $\: \: \: \: \: \: \: \: \: \: \:{\pink{\frak{As \: we \: know \: that,}}}$

Total Surface Area (cylinder) is the sum of it's Curved Surface Area and the base area and the top of cylinder.

That's why,

$\; \; \; \; \; \;{\bold{\bf{\longmapsto TSA \: = \: CSA + 2 \pi r^{2}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto TSA = \: \dfrac{TSA}{3} + 2 \pi r^{2}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto TSA - \dfrac{TSA}{3} = 2 \pi r^{2}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto \dfrac{3TSA \: - TSA}{3} = 2 \pi r^{2}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto \dfrac{3TSA \: - 1TSA}{3} = 2 \pi r^{2}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto \dfrac{2TSA}{3} = 2 \pi r^{2}}}}$

$\; \; \; \; \; \; \;{\bullet}$ Let's cross multiply

$\; \; \; \; \; \;{\bold{\bf{\longmapsto \dfrac{2TSA}{3 \times 2} = \pi r^{2}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto \dfrac{TSA}{3} = \pi r^{2}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto \dfrac{462}{3} = \dfrac{22}{7} \times r^{2}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto r^{2} = \dfrac{462}{3} \times \dfrac{7}{22}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto r^{2} = \dfrac{21}{3} \times 7}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto r^{2} = 7 \times 7}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto r^{2} = 7^{2}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto \sqrt{r^{2}} = \sqrt{7^{2}}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto 7 \: m}}}$

$\; \; \; \; \; \;{\bold{\bf{\leadsto 7 \: m \: is \: radius}}}$

${\pink{\frak{Henceforth \; 7 \; m \; is \; radius \; of \; cylinder}}}$

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~ Now let's find CSA of the cylinder

$\; \; \; \; \; \;{\bold{\bf{\longmapsto CSA = \dfrac{TSA}{3}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto CSA = \dfrac{462}{3}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto CSA = 154 \: m^{2}}}}$

${\pink{\frak{Henceforth \; CSA \; is \; 154m^{2} \; of \; cylinder}}}$

$\rule{150}{1}$

~ Now let's find the height of cylinder,

~ Finding height by CSA formula

CSA is given by,

$\; \; \; \; \; \;{\bold{\bf{\longmapsto CSA = 2 \pi rh}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto 154 = 2 \dfrac{22}{7} \times 7 \times h}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto 154 = \dfrac{44}{7} \times 7 \times h}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto 154 = 44 \times 7 \times h}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto \dfrac{154}{44} = h}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto \dfrac{7}{2} = h}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto h = \: \dfrac{7}{2}m}}}$

${\pink{\frak{Henceforth \; h \; is \; \dfrac{7}{2} m}}}$

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~ Now let's find the volume of cylinder

~ We have to use the formula to find volume of cylinder [ Put the values ]

~ Volume of cylinder is given by,

$\; \; \; \; \; \; \;{\boxed{\boxed{\red{Volume \; of \; cylinder = \pi r^{2}h}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto Volume \: = \: \pi r^{2}h}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto Volume \: = \: \dfrac{22}{7} \times 7^{2} \times \dfrac{7}{2}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto Volume \: = \: \dfrac{22}{7} \times 7 \times 7 \times \dfrac{7}{2}}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto Volume \: = \: \dfrac{11}{7} \times 7 \times 7 \times 7}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto 11 \times 7 \times 7}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto 11 \times 49}}}$

$\; \; \; \; \; \;{\bold{\bf{\longmapsto 539 \: m^{3}}}}$

${\pink{\frak{Henceforth \; v \: is \: 539m^{3}}}}$

$\rule{150}{1}$