Question

The radii of two cylinders are in the ratio 2:3 and their heights are in the ratio 5:3. Calculate the ratio of their curved surface areas.​

Answer 1

GivEn :

• Ratio of radii of the cylinder = 2 : 3

• Ratio of height of the cylinder = 5 : 3

To find :

• Curved Surface Area (CSA) of the cylinder = ?

Formula usEd :

Before proceeding to the question, let's understand formula of CSA of the cylinder !!

CSA is the area of only curved surface of the cylinder, i.e., leaving the top and base.

It is given by,

$\dag \large \: { \boxed{ \rm{CSA = 2\pi rh}}}$

where,

• r = radius of circular base and top

• h = height

⠀━━━━━━━━━━━━━━

SoluTion :

Let us assume that the radii of cylinder's be 2x and 3x , while height of the cylinder's be 5y and 3y .

According to the formula,

• 1st case [ CSA of 1st cylinder ]

$\sf : \implies \: \: \: CSA = 2\pi rh$

$\sf : \implies \: \: \: CSA = 2 \times \dfrac{22}{7} \times 2x \times 5y$

$\sf : \implies \: \: \: CSA = \dfrac{44}{7} \times 2x \times 5y$

⠀━━━━━━━━━━━━━━

• 2nd case [ CSA of 2nd cylinder]

$\sf : \implies \: \: \: CSA = 2\pi RH$

$\sf : \implies \: \: \: CSA = 2 \times \dfrac{22}{7} \times 3x \times 3y$

$\sf : \implies \: \: \: CSA = \dfrac{44}{7} \times 3x \times 3y$

⠀━━━━━━━━━━━━━━

★ Ratio of CSA of the 2 cylinders :

$\sf : \implies \: \: \: \dfrac{ \cancel\dfrac{44}{7} \times 2x \times 5y}{ \cancel\dfrac{44}{7} \times 3x \times 3y}$

$\sf : \implies \: \: \: \dfrac{ 2x \times 5y}{ 3x \times 3y}$

$\sf : \implies \: \: \: \dfrac{ 10 \cancel{xy}}{ 9\cancel{xy}}$

$\sf : \implies \: \: \: \dfrac{ 10}{ 9}$

$\sf : \implies \: \: \: { \underline{ \boxed{ \sf{ \purple{ \: 10 :9 \: }}}}} \ \bigstar$

Hence, ratio of CSA of cylinder is 10 : 9 .

⠀━━━━━━━━━━━━━━

Answer 2

Answer:

GivEn :

Ratio of radii of the cylinder = 2 : 3

Ratio of height of the cylinder = 5 : 3

To find :

Curved Surface Area (CSA) of the cylinder = ?

Formula usEd :

Before proceeding to the question, let's understand formula of CSA of the cylinder !!

CSA is the area of only curved surface of the cylinder, i.e., leaving the top and base.

It is given by,

\dag \large \: { \boxed{ \rm{CSA = 2\pi rh}}}†

CSA=2πrh

where,

r = radius of circular base and top

h = height

⠀━━━━━━━━━━━━━━

SoluTion :

Let us assume that the radii of cylinder's be 2x and 3x , while height of the cylinder's be 5y and 3y .

According to the formula,

1st case [ CSA of 1st cylinder ]

\begin{gathered}\sf : \implies \: \: \: CSA = 2\pi rh \\ \\ \\ \\\end{gathered}

:⟹CSA=2πrh

\begin{gathered}\sf : \implies \: \: \: CSA = 2 \times \dfrac{22}{7} \times 2x \times 5y \\ \\ \\ \\\end{gathered}

:⟹CSA=2×

7

22

×2x×5y

\begin{gathered}\sf : \implies \: \: \: CSA = \dfrac{44}{7} \times 2x \times 5y \\ \\\end{gathered}

:⟹CSA=

7

44

×2x×5y

⠀━━━━━━━━━━━━━━

2nd case [ CSA of 2nd cylinder]

\begin{gathered}\sf : \implies \: \: \: CSA = 2\pi RH \\ \\ \\ \\\end{gathered}

:⟹CSA=2πRH

\begin{gathered}\sf : \implies \: \: \: CSA = 2 \times \dfrac{22}{7} \times 3x \times 3y \\ \\ \\ \\\end{gathered}

:⟹CSA=2×

7

22

×3x×3y

\begin{gathered}\sf : \implies \: \: \: CSA = \dfrac{44}{7} \times 3x \times 3y \\ \\\end{gathered}

:⟹CSA=

7

44

×3x×3y

⠀━━━━━━━━━━━━━━

★ Ratio of CSA of the 2 cylinders :

\begin{gathered}\sf : \implies \: \: \: \dfrac{ \cancel\dfrac{44}{7} \times 2x \times 5y}{ \cancel\dfrac{44}{7} \times 3x \times 3y} \\ \\ \\ \\\end{gathered}

:⟹

7

44

×3x×3y

7

44

×2x×5y

\begin{gathered}\sf : \implies \: \: \: \dfrac{ 2x \times 5y}{ 3x \times 3y} \\ \\ \\ \\\end{gathered}

:⟹

3x×3y

2x×5y

\begin{gathered}\sf : \implies \: \: \: \dfrac{ 10 \cancel{xy}}{ 9\cancel{xy}} \\ \\ \\ \\\end{gathered}

:⟹

9

xy

10

xy

\begin{gathered}\sf : \implies \: \: \: \dfrac{ 10}{ 9} \\ \\ \\ \\\end{gathered}

:⟹

9

10

\begin{gathered}\sf : \implies \: \: \: { \underline{ \boxed{ \sf{ \purple{ \: 10 :9 \: }}}}} \ \bigstar \\ \\\end{gathered}

:⟹

10:9

★

Hence, ratio of CSA of cylinder is 10 : 9 .