Question

Height of circular cone is 3 times its diameter and volume is 216πcm×cm×cm. Find height of cone

Answer 1

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GIVEN :

DIMENSIONS OF A CONE ;

=> HEIGHT :

=> 3 * DIAMETER

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VOLUME :

=> 216 * π CM^3

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TO FIND :

=> THE HEIGHT OF THAT CONE ....

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FROM THE GIVEN INFO ,

=> WE SHOULD KNOW THE FORMULA OF VOLUME OF A CONE :

THAT'S :

=> 1 / 3 π R^2 H

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STEP 1 :

TO CONSIDER RADIUS , DIAMETER , HEIGHT

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CONSIDER ,

=> THE RADIUS AS X CM

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DIAMETER :

=> 2 * RADIUS

=> 2 * X

=> 2X CM

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HEIGHT :

=> 3 * DIAMETER

=> 3 * 2X

=> 6X CM

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STEP 2 :

TO MAKING THE EQUATION BY PUTTING THESE VALUES

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NOW PUT THESE VALUES IN THE FORMULA :

=> 1 / 3 * π * X^2 * 6X

=> π * 2X * X^2

=> π * 2X^3 CM^3

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BUT WE HAVE GIVEN THE VOLUME :

=> 216 * π CM^3

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STEP 3 :

TAKE THE VALUES AS EQUIVALENT

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=> 216 * π CM^3 = π * 2X^3 CM^3

=> 216 = 2X^3

=> 216 / 2 = X^3

=> 108 = X^3

=> X = 3√ 108

=> X = 4.76 CM

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RADIUS :

=> X CM

=> 4.76 CM

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STEP 4 :

WE HAVE GOT RADIUS ....

JUST FIND HEIGHT AS H = 6X

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HEIGHT :

=> 6X CM

=> 6 * 4.76 CM

=> 28.56 CM

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THANKS ....

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Answer 2

$\underline{\underline{\huge{\textbf{Solution:}}}}$

Given,
Height of the circular cone $h$ = 3(Diameter of the base)
Volume $v= 216\: cm^{3}$

$\underline{\large{\textsf{Step-1:}}}$
Express the diameter of the base of the circular cone in terms of radius

Since,
Shape of base of circular cone is a Circle

We have,
$\bigstar \mathbf{Diameter = 2 \times Radius}$

So,
Height of Circular cone = 3(2×Radius of the base)

Height of Circular cone = 6(Radius of the base)

$\underline{\large{\textsf{Step-2:}}}$
Assume a variable for the Radius of base of cone

Let $x\: cm$ be the radius of the base of circular cone

$\underline{\large{\textsf{Step-3:}}}$
Express the Height of circular cone in terms of radius of it's base.

Height of Circular cone $h = 6x\: cm$ --(1)

$\underline{\large{\textsf{Step-4:}}}$
Find the volume of circular cone

We have,
$\bigstar \mathbf{Volume\: of\: circular\: cone=\frac{1}{3}\, \pi r^{2} h}$

By Substituting,

Volume of circular cone $V = \frac{1}{3}\, \pi x^{2} (6x)$

$\implies V = \pi x^{2} (2x)$

$\implies V = 2 \pi x^{3}\: cm^{3}$

$\underline{\large{\textsf{Step-5:}}}$
Equating Volume of the cone to $216\pi\: cm^{3}$ to find value of x
(As both volumes refers to the same cone)

$\implies 2 \pi x^{3} = 216 \pi$

$\implies x^{3} = 108$

$\implies x = \sqrt[3]{108}$

$\implies x = 4.76$

$\therefore$
Radius of base of circular cone = 4.76 cm

$\underline{\large{\textsf{Step-6:}}}$
Find the Height of circular cone

From (1)
Height of circular cone h= 6(4.76) = 28.56 cm

$\therefore$

$\star \textsf{Height of the circular cone = \underline{\mathbf{28.56\: cm}}}$