Question

A right circular cone of radius 3 cm , has a curved surface area of 47.1 cm2 . Find the volume of the cone ( = 3.14)

Answer 1

Given:-

Radius of cone = 3cm

C.S.A of cone = 47.1sq.cm

To find

The volume of cone

Take π = 3.14

Now

A.T.Q

$\sf\underline{\dag{C.S.A\:of\:cone =\pi r l}}$

• Find the value of l

$\sf\implies \pi r l= 47.1\\ \\ \sf\implies 3.14\times 3\times l= 47.1\\ \\ \sf\implies 9.42\times l=47.1\\ \\ \sf\implies l=\cancel{\dfrac{47.1}{9.42}}\\ \\ \sf\implies l=5cm$

$\sf\underline{\dag{Volume\:of\:cone =\pi r{}^{2}h}}$

• Now find the value of h first

$\boxed{\sf{ l{}^{2}=h{}^{2}+r{}^{2}}}$

$\sf\implies (5){}^{2}= h{}^{2}+(3){}^{2}\\ \\ \sf\implies 25=h{}^{2}+9\\ \\ \sf\implies h{}^{2}=25-9\\ \\ \sf\implies h{}^{2}=16\\ \\ \sf\implies h=\sqrt{16}\\ \\ \sf\implies h=4cm$

So finally find the volume of cone

$\sf\implies Volume\:of\:cone =\dfrac{1}{3} \pi r{}^{2} h\\ \\ \sf\implies Volume= \dfrac{1}{\cancel3}\times 3.14\times 3\times \cancel{3}\times 4\\ \\ \sf\implies Volume= 3.14\times 3\times 4\\ \\ \sf\implies Volume =37.68cm{}^{3}$

$\sf{\therefore{ Volume\:of\:cone = 37.68cm{}^{3}}}$

Answer 2

Answer:

$\star \: \sf\green{Volume \: of \: cone} = {\boxed{\sf\blue{37.68 \: cm^3 }}}$

Solution:-

Given:-

• Radius of cone = 3 cm

• CSA of cone = 47.1 sq.cm

To find:-

• Volume of cone = ?

We know,

$\star{\boxed{\rm{CSA_{cone} = \pi rl }}}$

$\dashrightarrow\sf 47.1 = 3.14 \times 3 \times l \\\\\dashrightarrow\sf l = 47.1/(3 \times 3.14) \\\\\dashrightarrow\sf l = 47.1/9.42 \\\\\dashrightarrow\large{\underline{\boxed{\sf\blue{l = 5 \: cm }}}}$

$\rule{100}{2}$

For finding volume of cone,we must know value of height of cone.

Now,we know,

$\star{\boxed{\sf\green{Slant \: height(l)^2 = (Height)^2 + (Radius)^2 }}}$

$\dashrightarrow\sf (5)^2 = h^2 + (3)^2 \\\\\dashrightarrow\sf 25 = h^2 + 9 \\\\\dashrightarrow\sf h^2 = 25 - 9 \\\\\dashrightarrow\sf h^2 = 16 \\\\\dashrightarrow\sf h =\sqrt{16} \\\\\dashrightarrow\large{\underline{\boxed{\sf\green{h = 4 \: cm }}}}$

$\rule{100}{2}$

We also know that,

$\star{\boxed{\sf\red{Volume_{cone} = \dfrac{1}{3} \pi r^2 h }}}$

$\dashrightarrow\sf Volume_{cone} = \dfrac{1}{3} \times 3.14 \times (3)^2 \times 4 \\\\\dashrightarrow\sf Volume_{cone} = \dfrac{1}{3}\times 113.04 \\\\\dashrightarrow\large{\underline{\boxed{\sf\blue{Volume_{cone} = 37.68 \: cm^3 }}}}$