Question

The total surface area of a cylinder is
66pi cm². If its height is 8 cm, find the base
radius of the cylinder.​

Answer 1

Answer:

• r = 3 cm

Given:

• Total Surface Area = 66π cm²
• Height of the cylinder = 8 cm

To Find:

• Radius of the cylinder = ?

Steps:

Formula to calculate the Total Surface Area of a cylinder is given as:

$\boxed{ \text{TSA of cylinder} = 2 \pi r ( r + h )}$

Substituting the given information in the formula we get:

$\implies 66 \pi = 2\pi r ( 8 + r )\\\\\\\text{Transposing 2} \: \pi\: \text{to the LHS we get,}\\\\\\\implies \dfrac{ 66 \pi}{2 \pi} = r ( 8 + r )\\\\\\\implies 33 = 8r + r^2\\\\\\\implies r^2 + 8r - 33 = 0\\\\\\\text{Solving this equation we get,}$

$\implies r^2 + 11r - 3r - 33 = 0\\\\\\\implies r ( r +11 ) - 3 ( r + 11 ) = 0\\\\\\\implies ( r + 11 ) ( r -3 ) = 0\\\\\\\implies r = (-11) \:\: and \:\: 3\\\\\text{But since 'r' cannot be negative we eliminate}\:\:-11.\:\:\text{Hence we get,}\\\\\implies \boxed{\bf{r = 3\:cm}}$

Hence the radius of the base of the cylinder is 3 cm.

Answer 2

$\LARGE\textbf{\underline{\underline{Given :-}}}$

$\large\texttt{↦ T.S.A. of the cylinder = 66 π cm² . }\\\\\large\texttt{↦ Height of the cylinder = 8 cm .}$

$\LARGE\textbf{\underline{\underline{To find :-}}}$

$\large\texttt{↦ The base of the cylinder = ?}\\\\\large\texttt{↦ The radius of the cylinder = ?}$

$\LARGE\textbf{\underline{\underline{Formula :-}}}$

$\large \underline{\boxed{\textsf\textcolor{green}{➙ T.S.A. _ { ( \: Cylinder \: ) } = 2\pi r \: ( r + h ) }}}$

$\LARGE\textbf{\underline{\underline{Solution :-}}}$

$\large\: \bigstar\textsf\textcolor{orange}{\: \: \: T.S.A. _ { ( \: Cylinder \: ) } = 2\pi r \: ( r + h ) }\\\\\\\large: \: \Longrightarrow\textsf{ 66 \: \pi = 2 \pi r \: ( r + 18) }\\\\\\\large: \: \Longrightarrow\textsf{ \cfrac{66 × \pi}{\pi} = 2r \: ( r + 8 ) }\\\\\\\large: \: \Longrightarrow\textsf{ \cfrac{66 × \cancel\pi}{\cancel\pi} = 2r^{2} + 16r }\\\\\\\large: \: \Longrightarrow\textsf{- 2r² - 16r + 66 = 0 .....[ Divide by 2 on both the equation]}\\\\\\\large: \: \Longrightarrow\underline{\boxed{\textsf\textcolor{purple}{- r² - 8r + 33 = 0}}}$

$\large:\: \bigstar\textsf\textcolor{orange}{\: \: \: By factorisation method :-}\\\\\\\large: \: \Longrightarrow\textsf{- r² - 8r + 33 = 0}\\\\\\\large: \: \Longrightarrow\textsf{- r² - 11r + 3r + 33 = 0 }\\\\\\\large: \: \Longrightarrow\textsf{- r ( r + 11 ) + 3 ( r + 11 ) = 0}\\\\\\\large: \: \Longrightarrow\underline{\boxed{\textsf\textcolor{purple}{ ( r + 11 ) ( - r + 3 ) = 0 }}}$

➙ Now :-

$\large: \: \Longrightarrow\textsf{r + 11 = 0 \: \: \: \: or \: \: \: \: - r + 3 = 0}\\\\\\\large: \: \Longrightarrow\textsf{r = - 11 \: \: \: \: or \: \: \: \: - r = 3 }\\\\\\\large : \: \Longrightarrow\underline{\boxed{\textsf\textcolor{purple}{r = - 11 \: \: \: \: or \: \: \: \: r = 3 }}}$

• Now radius can't be negative .

• So , the radius of the cylinder = 3 cm .

$\large : \: \Longrightarrow\underline{\boxed{\textsf\textcolor{purple}{ Radius _ { ( \: Cylinder \: ) } = 3 \: cm . }}}$