Question

The volume of a right circular cone is 9856 cm. If the diameter of the base is 28 cm, find
(i) height of the cone
(ii) slant height of the cone
(iii) curved surface area of the cone
​

Answer 1

Given :

• Volume of a right circular cone = 9856 cm³
• Diameter of the base = 28 cm

To find :

• Height of the cone
• Slant height of the cone
• Curved surface area of the cone

Concept :

➳ Formula to calculate radius :-

$\boxed{ \pmb{ \sf{Radius = \dfrac{Diameter}{2}}}}$

➳ Formula of volume of a right circular cone :-

$\boxed{ \pmb{ \sf{ Volume \: of \: the \: right \: circular \: cone = \pi {r}^{2} \dfrac{h}{3}}}}$

➳ Formula of slant height of the cone :-

$\boxed{ \pmb{ \sf{ Slant \: height\: of \: the \: right \: circular \: cone = \sqrt{ {r}^{2} + {h}^{2}}}}}$

➳ Formula of curved surface area of the cone :-

$\boxed{ \pmb{ \sf{ Curved \: surface \: area\: of \: the \: right \: circular \: cone= \pi rl }}}$

where,

• Take π = 22/7
• r = radius of the cone
• h = height of the cone
• l = slant height of the cone

Solution :

Firstly, calculate the radius of the cone.

$\\ \dashrightarrow \quad \sf Radius = \dfrac{Diameter}{2}$

$\\ \dashrightarrow \quad \sf Radius = \dfrac{28}{2}$

$\\ \dashrightarrow \quad \sf Radius = \dfrac{ \not28}{ \not2}$

$\\ \dashrightarrow \quad \sf Radius = 14 \: cm$

Height of the cone :-

$\\ \dashrightarrow \sf Volume \: of \: a \: right \: circular \: cone = \pi {r}^{2} \dfrac{h}{3}$

$\\ \dashrightarrow \sf \quad 9856= \dfrac{22}{7} \times {(14)}^{2} \times \dfrac{h}{3}$

$\\ \dashrightarrow \sf \quad 9856= \dfrac{22}{7} \times 14 \times 14 \times \dfrac{h}{3}$

$\\ \dashrightarrow \sf \quad 9856= \dfrac{22}{ \not7} \times 14 \times \not14 \times \dfrac{h}{3}$

$\\ \dashrightarrow \sf \quad 9856= 22 \times 14 \times 2 \times \dfrac{h}{3}$

$\\ \dashrightarrow \sf \quad \dfrac{9856 \times 3}{22 \times 14 \times 2} = h$

$\\ \dashrightarrow \sf \quad 48 = h$

• Height of the cone = 48 cm

Slant height of the cone :-

$\\ \dashrightarrow \sf \quad Slant \: height\: of \: a \: right \: circular \: cone = \sqrt{ {r}^{2} + {48}^{2}}$

$\\ \dashrightarrow \sf \quad Slant \: height = \sqrt{ {14}^{2} + {48}^{2}}$

$\\ \dashrightarrow \sf \quad Slant \: height = \sqrt{ 196 + 2304}$

$\\ \dashrightarrow \sf \quad Slant \: height = \sqrt{2500}$

$\\ \dashrightarrow \sf \quad Slant \: height = 50$

• Slant height of the cone = 50 cm

Curved surface area of the cone :-

$\\ \dashrightarrow\quad \sf Curved \: surface \: area\: of \: right \: circular \: cone= \pi rl$

$\\ \dashrightarrow\quad \sf CSA \: of \: cone = \dfrac{22}{7} \times 14 \times 50$

$\\ \dashrightarrow\quad \sf CSA \: of \: cone = \dfrac{22}{ \not7} \times \not14 \times 50$

$\\ \dashrightarrow\quad \sf CSA \: of \: cone = 22 \times 2 \times 50$

$\\ \dashrightarrow\quad \sf CSA \: of \: cone = 2200$

• Curved surface area of the cone = 2200 cm²

Answer 2

Answer:

Given :-

• The volume of a right circular cone is 9856 cm.
• Diameter of the base is 28 cm.

To Find :-

• (i) Height of the cone
• (ii) Slant height of the cone
• (iii) Curved surface area of the cone

Solution :-

${\large{\bold{\purple{\underline{1)\: Height\: Of\: The\: Cone\: :-}}}}}$

First, we have to find the radius of the cone :

As we know that :

$\longmapsto \sf\boxed{\bold{\pink{Radius =\: \dfrac{Diameter}{2}}}}$

Given :

• Diameter = 28 cm

Then,

$\implies \sf Radius =\: \dfrac{\cancel{28}}{\cancel{2}}$

$\implies \sf\bold{\green{Radius =\: 14\: cm}}$

Hence, the radius of the cone is 14 cm .

Now, we have to find the height of the cone :

Let, consider height of the cone be h cm.

As we know that :

$\longmapsto \sf\boxed{\bold{\pink{Volume\: Of\: Cone =\: \dfrac{1}{3}{\pi}{r}^{2}h}}}$

where,

• r = Radius
• h = Height

Given :

• Radius = 14 cm

According to the question by using the formula we get :

$\implies \sf \dfrac{1}{3}{\pi}{r}^{2}h =\: 9856$

[ We know that : π = 22/7 ]

$\implies \sf \dfrac{1}{3} \times \dfrac{22}{7} \times {(14)}^{2} \times h =\: 9856$

$\implies \sf \dfrac{1}{3} \times \dfrac{22}{7} \times 196 \times h =\: 9856$

$\implies \sf h =\: \dfrac{9856 \times 3 \times 7}{22 \times 196}$

$\implies \sf h =\: \dfrac{\cancel{206976}}{\cancel{4312}}$

$\implies \sf\bold{\red{h =\: 48\: cm}}$

$\therefore$ The height of the cone is 48 cm .

$\rule{300}{2}$

${\large{\bold{\purple{\underline{2)\: Slant\: height\: Of\: The\: Cone\: :-}}}}}$

As we know that :

$\longmapsto \sf\boxed{\bold{\pink{Slant\: Height\: Of\: Cone =\: \sqrt{{r}^{2} + {h}^{2}}}}}$

where,

• r = Radius
• h = Height

Given :

• Height = 48 cm
• Radius = 14 cm

According to the question by using the formula we get :

$\implies \sf Slant\: height\: (l) =\: \sqrt{{14}^{2} + {48}^{2}}$

$\implies \sf Slant\: height\: (l) =\: \sqrt{196 + 2304}$

$\implies \sf Slant\: height\: (l) =\: \sqrt{2500}$

$\implies \sf\bold{\red{Slant\: height\: (l) =\: 50\: cm}}$

$\therefore$ The slant height of the cone is 50 cm .

$\rule{300}{2}$

${\large{\bold{\purple{\underline{3)\: Curved\: Surface\: Area\: Of\: The\: Cone\: :-}}}}}$

As we know that :

$\implies \sf\boxed{\bold{\pink{Curved\: Surface\: Area\: Of\: Cone =\: {\pi}rl}}}$

where,

• r = Radius
• l = Slant height

Given :

• Radius = 14 cm
• Slant height = 50 cm

According to the question by using the formula we get,

$\implies \sf C.S.A\: Of\: Cone =\: \dfrac{22}{7} \times 14 \times 50$

$\implies \sf C.S.A\: Of\: Cone =\: \dfrac{22}{\cancel{7}} \times {\cancel{700}}$

$\implies \sf C.S.A\: Of\: Cone =\: 22 \times 100$

$\implies \sf\bold{\red{C.S.A\: Of\: Cone =\: 2200\: {cm}^{2}}}$

$\therefore$ The curved surface area or CSA of the cone is 2200 cm².

$\rule{300}{2}$