Question

if the circumference of the base of cylinder is 44cm and the sum of its radius and height is 27 cm, find its total surface area.​

Answer 1

$\sf Given \begin{cases} & \sf{Circumference\:of\:the\:base\;of\:cylinder = \bf{44\:cm}} \\ & \sf{Sum\:of\:radius\:and\:height\:of\:cylinder = \bf{27\:cm}} \end{cases}$

To find: Total surface area of cylinder?

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☯ Let's consider r and h be the radius and height of cylinder respectively.

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

$\star\;{\boxed{\sf{\pink{Circumference_{\;(circle)} = 2 \pi r}}}}$

$:\implies\sf 2 \times \dfrac{22}{7} \times r = 44$

$:\implies\sf \dfrac{44}{7} \times r = 44$

$:\implies\sf r = \cancel{44} \times \dfrac{7}{ \cancel{44}}$

$:\implies{\underline{\boxed{\frak{\purple{r = 7\:cm}}}}}\;\bigstar$

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

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

• Sum of radius and height of cylinder is 27 cm.

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$:\implies\sf r + h = 27$

$:\implies\sf 7 + h = 27$

$:\implies\sf h = 27 - 7$

$:\implies{\underline{\boxed{\frak{\purple{h = 20\:cm}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Height\:of\:cylinder\:is\: {\textsf{\textbf{20\:cm}}}.}}}$

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☯ Now, Finding Curved surface area of cylinder,

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$\star\;{\boxed{\sf{\pink{Total\:surface\:area_{\;(rectangle)} = 2 \pi r(r + h)}}}}$

$:\implies\sf 2 \times \dfrac{22}{ \cancel{7}} \times \cancel{7} \bigg( 7 + 20 \bigg)$

$:\implies\sf 2 \times 22 \times 27$

$:\implies{\underline{\boxed{\frak{\purple{1188\:cm^2}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Total\:surface\:area\:of\:cylinder\:is\: \bf{1188\:cm^2}.}}}$

Answer 2

$\begin{gathered}{\underline{\underline{\maltese{\large{\textsf{\textbf{\red{ Given:}}}}}}}}\end{gathered}$

➲ Circumference of the base of Cylinder = 44 cm

➲ Sum of radius and height if Cylinder = 27 cm

$\begin{gathered}{\underline{\underline{\maltese{\large{\textsf{\textbf{\red{ To Find:}}}}}}}}\end{gathered}$

➲Total surface area

$\begin{gathered}{\underline{\underline{\maltese{\large{\textsf{\textbf{\red{ Using Formula:}}}}}}}}\end{gathered}$

${\dag{\underline{\boxed{\sf{Circumference \: of \: Cylinder =2{\pi}r}}}}}$

$\dag{\underline{\boxed{\sf{Total \: surface \: area \: of \: Cylinder = 2{\pi}r(h + r), }}}}$

$\begin{gathered}{\underline{\underline{\maltese{\large{\textsf{\textbf{\red{ Solution:}}}}}}}}\end{gathered}$

${\dag{\underline{\underline{\frak{\green{Let \: the \: : }}}}}}$

• Radius of Cylinder = r
• Height of Cylinder = h

${\dag{\underline{\underline{\frak{\green{Finding \: the \: radius \: of \: Cylinder. \: : }}}}}}$

According to the question,

$\quad{: \implies{\sf{Circumference \: of \: Cylinder =2{\pi}r}}}$

Substituting the values

$\begin{gathered} \begin{gathered}\quad{: \implies{\sf{44 =2 \times {\dfrac{22}{7}} \times r}}} \\ \\ \quad{: \implies{\sf{44 ={\dfrac{44}{7}} \times r}}} \\ \\ \qquad{: \implies{\sf{44 \times {\dfrac{7}{44}} = r}}} \\ \\ \qquad{: \implies{\sf{\cancel{44} \times {\dfrac{7}{\cancel{44}}} = r}}} \\ \\ \qquad{: \implies{\sf{7 = r}}} \\ \\\qquad\dag{\underline{\boxed{\sf{\purple{r = 7\: cm}}}}}\end{gathered}\end{gathered}$

${\dag{\underline{\underline{\frak{\green{Finding \: the \: height \: of \: Cylinder \: : }}}}}}$

According to the question,

$\quad{: \implies{\sf{radius + height = 27 cm}}}$

Substituting the values

$\begin{gathered} \begin{gathered}\quad{: \implies{\sf{7 + height = 27 cm}}} \\ \\ : \implies{\sf{ height = 27 - 7}} \\ \\ : \implies{\sf{ height = 20 \: cm}} \\ \\ \dag{\underline{\boxed{\sf{\purple{ height = 20 \: cm}}}}}\end{gathered}\end{gathered}$

${\dag{\underline{\underline{\frak{\green{Now \: Finding \: the \: total \: surface \: area \: of \: Cylinder \: : }}}}}}$

$\begin{gathered} \quad{ : \implies{\sf{Total \: surface \: area \: of \: Cylinder = 2{\pi}r(h + r)}}} \end{gathered}$

Substituting the values

$\begin{gathered} \begin{gathered} \quad{ : \implies{\sf{Total \: surface \: area \: of \: Cylinder = 2 \times { \dfrac{22}{7}} \times 7(20 + 7)}}} \\ \\ \qquad{: \implies{\sf{Total \: surface \: area \: of \: Cylinder = 2 \times { \dfrac{22}{7}} \times 7(27)}}} \\ \\ \qquad{: \implies{\sf{Total \: surface \: area \: of \: Cylinder = 2 \times { \dfrac{22}{7}} \times 7 \times 27}}} \\ \\ \qquad{: \implies{\sf{Total \: surface \: area \: of \: Cylinder = 2 \times { \dfrac{22}{\cancel{7}}} \times \cancel{7} \times 27}}} \\ \\ \qquad{: \implies{\sf{Total \: surface \: area \: of \: Cylinder = 2 \times 22 \times 27}}} \\ \\ \qquad{: \implies{\sf{Total \: surface \: area \: of \: Cylinder = 1188 \: {cm}^{2}}}} \\ \\ \qquad\dag{ \underline{\boxed{\sf{\pink{Total \: surface \: area \: of \: Cylinder = 1188 \: {cm}^{2}}}}}}\end{gathered}\end{gathered}$