Question

- A solid cylinder has a total surface area of 462 sq. cm. Its curved surface area is one-third of its total surface area. Find the volume of the cylinder.

Answer 1

Given : A solid cylinder has a total surface area of 462 sq. cm and It's curved surface area is one-third of its total surface area .

Need To Find : Volume of Cylinder.

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

❍ Let's consider the Radius of Cylinder be r and height of Cylinder be h .

$\frak{\underline { As,\:We\:know\:that\::}}$

⠀⠀⠀⠀⠀ $\red {\underline {\sf{ Total \:Surface\:Area _{(Cylinder)} = 2 \times \pi \times Radius ( Height + Radius) \:sq.\:units .}}}$

⠀⠀⠀⠀⠀ $\red {\underline {\sf{ Curved \:Surface\:Area _{(Cylinder)} = 2 \times \pi \times Radius \times Height \:sq.\:units .}}}$

Given that :

⠀⠀⠀⠀⠀ It's curved surface area is one-third of its total surface area .

Then :

⠀⠀⠀⠀⠀ $\pink {\underline {\sf{ Curved \:Surface\:Area _{(Cylinder)} = \dfrac{1}{3} (Total \:Surface\;Area_{(Cylinder)}).}}}$

Or ,

⠀⠀⠀⠀⠀ $\pink {\underline {\sf{ 2 \times \pi \times Radius \times Height = \dfrac{1}{3}\big( 2 \times \pi \times Radius ( Height + Radius)\big)}}}$

⠀⠀⠀⠀⠀⠀$\underline {\sf{\bf{\star\:Now \: By \: Substituting \: the \: Assumed \: Values \::}}}$

⠀⠀⠀⠀⠀ $:\implies {\sf{= ( 2 \times \pi \times r \times h)=\dfrac{1}{3}\big( 2 \times \pi \times r ( h + r) \big) }}$

⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{3 \times 2 \times \pi \times r \times h = 2 \times \pi \times r ( h + r) }}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{6 \times \pi \times r \times h = 2 \times \pi \times r (h + r ) }}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{6 \times \pi \times r \times h = 2 \times \pi \times r (h + r) }}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{6 \times \pi \times r \times h = 2 \times \pi\times r\times h + 2\times \pi \times r^{2} }}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{6 \times \pi \times r \times h- 2 \times \pi\times r\times h = 2\times \pi \times r^{2} }}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{\cancel {4 } \times \pi \times r \times h = \cancel {2}\times \pi \times r^{2} }}$

⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{2 \times \cancel {\pi} \times r \times h = \cancel {\pi} \times r^{2} }}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{2 \times \cancel {r} \times h = \cancel {r^{2}} = \: r }}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{2 \times h = \: r }}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀ $:\underline{\pink{\boxed{\sf{ r =\: 2h }}}}\quad \bf{\bigstar}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

Given That ,

• A solid cylinder has a total surface area of 462 sq. cm.

$\frak{\underline { As,\:We\:know\:that\::}}$

• ⠀⠀⠀ ⠀⠀⠀⠀⠀ It's curved surface area is one-third of its total surface area .

• Curved surface area = $\dfrac{1}{3} \times 462$

• Curved surface area = $\dfrac{1}{\cancel {3}} \times \cancel{462}$

• Curved surface area = 154 cm²

Or ,

• $2\times \pi\times\:r \times h$ = 154 cm²

• ⠀⠀Here : $:\implies {\sf{2 \times h = \: r }}$

⠀⠀⠀⠀⠀ $:\implies {\sf{2 \times \dfrac{22}{7} \times 2h \times h = 154 cm^{2} }}$

⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{2 \times \dfrac{22}{7} \times 2h^{2} = 154 cm^{2} }}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{ h^{2} = \dfrac{154\times 7 }{22 \times 2 \times 2 } }}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{ h^{2} = \dfrac{\cancel {1,078}}{\cancel {88} } }}$

⠀⠀⠀⠀⠀ ⠀⠀⠀⠀$:\implies {\sf{ h^{2} = \dfrac{49}{4} }}$

⠀⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{ h =\sqrt { \dfrac{49}{4}} }}$

⠀⠀⠀⠀⠀ $:\underline {\pink{\sf{\boxed{ h = \dfrac{7}{2} cm }}}}\quad \bf{\bigstar}$

As , We know that ,

⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{2 \times h = \: r }}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{\cancel {2} \times \dfrac{7}{\cancel{2}} = \: r }}$

⠀⠀⠀⠀⠀ $:\underline {\pink{\sf{\boxed { r = 7 cm }}}}\quad \bf{\bigstar}$

Therefore,

• Radius of Cylinder is $: {\sf{ h = \dfrac{7}{2}\:cm }}$

• Height of Cylinder is $: {\sf{ h = \: 7\:cm }}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

As, We know that ,

⠀⠀⠀⠀⠀ $\red{\underline {\sf{ Volume _{(Cylinder)} = 2 \times \pi \times Radius^{2}\times Height \:cu.\:units .}}}$

⠀⠀⠀⠀⠀⠀$\underline {\sf{\bf{\star\:Now \: By \: Substituting \: the \: Found \: Values \::}}}$

⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{Volume _{(Cylinder)}\dfrac{22}{7} \times 7^{2} \times \dfrac{7}{2} }}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{Volume_{(Cylinder)} = \dfrac{22}{\cancel {7}} \times \cancel {7}\times 7 \times \dfrac{7}{2} }}$

⠀⠀⠀⠀⠀⠀⠀ $:\implies {\sf{Volume_{(Cylinder)} = \dfrac{1,078}{\cancel {2}} }}$

⠀⠀⠀⠀⠀ $:\underline {\pink{\boxed{\sf{Volume _{(Cylinder)} = 539cm^{2} }}}}$

Therefore,

⠀⠀⠀⠀⠀$\underline {\therefore\:{\pink{ \mathrm {  Hence \:, Volume \:of\:Cylinder \:is\:539\: cm^{2}}}}}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

Answer 2

Given,

• TSA of Cylinder = 462 sq. cm
• CSA is One - Third Of TSA.

To Find,

• The Volume of Cylinder.

Solution,

→ ⅓ of TSA of Cylinder = CSA of Cylinder

→ ⅓ × 462 sq. cm = CSA of Cylinder

→ 154 sq. cm = CSA of Cylinder

→ 154 sq. cm = CSA of Cylinder

TSA = Area Of Two Circles + CSA

→ 462 sq. cm - 154 sq. cm = Area Of Two Circles

→ 308 sq. cm = Area Of Two Circles

Then,

The Area Of One Circle = 154 sq. cm

→ Area of Circle = 154 sq. cm

→ π r² = 154 sq. cm

→ 22/7 × r² = 154 sq. cm

→ r² = 154 sq. cm × 7/22

→ r² = 49 sq. cm

→ r = 7cm

CSA of Cylinder = 154 sq. cm

→ 2 π r h = 154 sq. cm

→ 2 × 22/7 × 7cm × h = 154 sq. cm

→ 44cm × h = 154 sq. cm

→ h = 154 sq. cm /44cm

→ h = 7/2 cm

Required Answer,

The Volume of Cylinder = π r² h

→ 22/7 × (7cm)² × 7/2cm

→ 22/7 × 49cm² × 7/2cm

→ 539cm³