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

$\large\bf{\underline\orange{Answer \:↝}}$

Copper sphere radius r = 3 cm

Cone height h =3 cm

Radius of the base of cone R = ?

Volume of copper sphere = 4/3πr³

= 4/3π(3)³

= 36π cm³

Volume of Cone = 1/3πR²h

= 1/3πR²(3)

= πR²cm³

Volume of copper sphere = Volume of cone

∴ 36π=πR²

∴ R² =36

∴ R=6 cm