Question

the volume of a cylinder having height twice the radius is 17248 cm^3. what minimum square cm paper is needed to cover the curved surface area of the cylinder.

Answer 1

$\huge{\underline{\mathfrak{\textcolor{blue}{Answer :}}}}$ $\Large{\underline{\boxed{2464 \: {cm}^{2} }}}$
$\huge{\underline{\mathrm{\textcolor{red}{Step-by-step \: \: explanation :}}}}$

$\LARGE{\underline{\mathtt{\textcolor{violet}{Given :-}}}}$
⚪ Volume of cylinder, $\sf v = 17248 \: {cm}^{3}$
⚪ ( Height of cylinder ) = 2 ( radius of cylinder ) , i.e.: $\boxed{h = 2r}$

$\LARGE{\underline{\mathtt{\textcolor{green}{To \: \: find :-}}}}$
⚪ Minimum CSA

$\LARGE{\underline{\mathtt{\textcolor{teal}{Concept \: \: used :-}}}}$
⏺ Volume & Surface area of Solids ⏺

$\LARGE{\underline{\mathtt{\textcolor{blue}{Solution :-}}}}$

✒ let r and h be the radius and height of the cylinder respectively.

Therefore, h = 2r

Now,
$\rm \qquad volume = 17248 \: {cm}^{3} \\ \\ \implies\pi {r}^{2} h = 17248 \\ \\ \implies\pi \times r \times r \times 2r = 17248 \qquad( \sf \because \: h = 2r) \\ \\ \implies2\pi {r}^{3} = 17248 \\ \\ \implies 2 \times \frac{22}{7} \times {r}^{3} = 17248 \\ \\ \implies {r}^{3} = \frac{ {}^{784} \cancel{17248} \times 7}{ \cancel{22} \times 2} \\ \\ \qquad \quad \: \: \: = \frac{ {}^{392} \cancel{784} \times 7}{ \cancel{2}} \\ \\ \qquad \quad \: \: \: = 392 \times 7 \\ \\ \implies {r}^{3} = 2744 \\ \\ \implies \sqrt[3]{ {r}^{3} } = \sqrt[3]{2744} \\ \\ \implies \: r = 14 \: \sf \: cm$

Thus,
$\qquad \: h = 2r \\ \implies \: h = 2(14 \: cm) \\ \implies \: h = 28 \: cm$

Hence, CSA
$= 2\pi rh \\ = 2 \times \frac{22}{7} \times 14 \times 28 \\ = \frac{2 \times 22 \times 14 \times \cancel{28} \: {}^{4} }{ \cancel{7}} \\ = 44 \times 56 \\ = 2464 \: {cm}^{2}$

$\LARGE{\underline{\mathtt{\textcolor{magenta}{Conclusion :-}}}}$
⚪ Minimum square cm of paper needed is $2464 \: \sf {cm}^{2}$ .

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