Question

a copper sphere of diameter 6 cm is melted and recast into a right circular cone of radius 6 cm. then the height of the cone is​

Answer 1

Given :

• Diameter of the copper sphere = 6 cm
• Radius of the recast right circular cone = 6 cm

To Find :

• The height of the recast right circular cone

Solution :

Radius of sphere will be 6/3 cm = 2 cm. {Since , 2(radius) = diameter}

Here , In this case ;

• Volume of the Sphere will be equal to the volume of recast right circular cone

Volume of sphere is given by ,

$\\ \star \: {\boxed{\purple{\sf{Volume_{(sphere)} = \frac{4}{3}\pi {r}^{3} }}}}$

Here ,

• r is radius of the sphere

Substituting the values we have ,

$\\ : \implies \sf \: Volume_{(sphere)} = \frac{4}{3} \times \frac{22}{7} \times {(3)}^{3}$

$\\ : \implies \sf \: Volume_{(sphere)} = \frac{4}{3} \times \frac{22}{7} \times 27$

$\\ : \implies \sf \: Volume_{(sphere)} = \frac{4 \times 22 \times 9}{7}$

$\\ : \implies{\underline{\boxed{\red {\mathfrak{ Volume_{(sphere)} = \frac{792}{7} }}}}} \:$

$\qquad$━━━━━━━━━━━━━━━━━

Volume of a cone is given by ,

$\\ \star \: {\boxed{\purple{\sf{Volume_{(cone)} = \frac{1}{3}\pi {r}^{2}h }}}}$

Here ,

• r is radius of cone
• h is height of cone

Substituting the values we have ,

$\\ : \implies \sf \: Volume_{(cone)} = \frac{1}{3} \times \frac{22}{7} \times {(6)}^{2} \times h$

$\\ : \implies \sf \: Volume_{(cone)} = \frac{1}{3} \times \frac{22}{7} \times 36 \times h$

$\\ : \implies \sf \: Volume_{(cone)} = \frac{22 \times 12 \times h}{7}$

$\\ : \implies{\underline{\boxed{\red {\mathfrak{Volume_{(cone)} = \frac{264h}{7} }}}}}$

$\qquad$━━━━━━━━━━━━━━━━━

Now , Applying the condition ;

$\\ : \implies \sf \: \frac{792}{7} = \frac{264h}{7}$

$\\ : \implies \sf \: 792 = 264h$

$\\ : \implies \sf \: h = \frac{792}{264}$

$\\ : \implies{\underline{\boxed{\pink {\mathfrak{h = 3 \: cm}}}}} \: \bigstar$

$\\ \therefore{\underline{\sf{Hence \: , \: The \: height \: of \: the \: right \: \: circular \: cone \: is \: \bold{ 3 cm}}}}$

Answer 2

Answer:

Given :-

• Diameter of Copper sphere = 6 cm
• Radius of the recast right circular cone = 6 cm

To Find :-

Height

Solution :-

Firstly let's find radius of copper wire

As we know that

$\bf \: Radius = \dfrac{ Diameter}2$

$\sf \implies Radius = \dfrac{6}{2}$

$\sf \implies \: Radius = 3 \: cm$

Now,

Let's find Volume of Sphere

As we know that

$\bf \pink {Volume = \dfrac{4}{3} \pi \: { r}^{3} }$

$\sf \implies \: Volume = \frac{4}{3} \times \frac{22}{7} \times {3}^{3}$

$\sf \implies \: Volume = \cancel \frac{4}{3} \times \frac{22}{7} \times \cancel{ 27}$

$\sf \implies \: Volume = \dfrac{4 \times 22 \times 9}{7}$

$\sf \implies \: Volume = \dfrac{792}{7} cm$

Now,

$\bf \red{Volume = \dfrac13 \pi {r}^{2} h}$

$\sf \implies Volume = \frac{1}{3} \times \frac{22}{7} \times {6}^{2} \times h$

$\sf \implies Volume = \frac{1}{3} \times \frac{22}{7} \times 36 \times h$

$\sf \implies Volume = \dfrac{22 \times 12 \times h}{7}$

$\sf \implies Volume =\dfrac{ 264 \: h}{7}$

$\dag{\mathfrak { \underline {According \: to \: Question}}}$

$\tt \: Volume \: of \: cone = Volume \: of \: sphere$

$\sf \: \dfrac{792}{7} = \dfrac{264h}{7}$

By cross multiplication

$\sf \: 792 \times 7 = 264h \times 7$

$\sf \: 5544= 1848h$

$\sf \: h = \cancel{ \dfrac{5544}{1848} }$

$\mathfrak{h = 3 \: cm}$

Hence :-

Height of cone is 3 cm