Question

a cone of height 24 cm and radius of base 6 cm is made up of modelling clay.a child reshapes it in the form of a sphere .find the radius of the sphere.i know the answer but plz give a proper explanation.

Answer 1

Volume of a cone = 1/3 × πr²h


Here, h = 24
r = 6

=> volume =
$\frac{1}{3} \times \pi {r}^{2} h \\ \\ = > \frac{1}{3} \times \pi \times {6}^{2} \times 24 \\ \\ = \frac{1}{3} \times \pi \times 36 \times 24 \\ \\ = > \pi \times 36 \times 8 \\ \\ 288\pi$


The volume of cone = 288π cm³


Now it's reshaped into a sphere. After reshaping, the volume would be the same, since it is completely reshaped.
=> Volume of sphere = 288π cm³

We know that volume of sphere = 4/3 × πr³

=> 4/3 × πr³ = 288π
$\frac{4}{3} \pi {r}^{3} = 288\pi \\ \\ \\ = > {r}^{3} = 288\pi \div \frac{4\pi}{3} \\ \\ = > {r}^{3} = 288\pi \times \frac{3}{4\pi} \\ \\ = > {r}^{3} = 72 \times 3 \\ \\ = > {r}^{ 3} = 216 \\ \\ = > r = \sqrt[3]{216} \\ \\ = > r = 6$



Answer :- Radius of sphere is 6 cm

Answer 2

$\sf {\large{Question: }}$

$\sf{ \green{A \: Cone \: of \: Height \: 24 \: cm \: and \: Radius \: of \: Base \: 6 \: cm}} \\ \sf{ \red{is \: made \: up \: of \: modelling \: clay \: a \: child \: Reshapes \: it}} \\ \sf{ \blue{in \: the \: form \: of \: a \: Sphere.}} \\ \sf{ \blue{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: find \: the \: radius \: of \: the \: sphere ?}}$

$\sf{ \large{Method \: of \: Solution:}}$

$\sf{ \red{Given}}$

$\implies{ \sf{ \blue{Height \: of \: the \: cone \: = 24cm}}} \\ \\ \implies{ \sf{ \green{ Radius \: \: of \: the \: cone \: = 6cm}}} \\ \\ \\ \implies{ \sf{ \red {Volume \: of \: the \: cone = \frac{1}{3} \times \frac{22}{7} \times ( {6}^{2} )\times 24}}} \\ \\ \\ \implies{ \sf{ \red {Volume \: of \: the \: cone = 905.14cm}}}$

$\sf{ \fbox{ \blue{According \: to \: the \: Question:}}}$

$\sf{\red{Volume \: of \: the \: Sphere \: = \frac{3}{4} \times \frac{22}{7} \times ({r}^{3})}}$

$\sf{\blue{Volume \: of \: the \: Sphere \: = \frac{3}{4} \times \frac{3}{4} \times ({r}^{3})=905.14}}$

$\sf{\green{Volume \: of \: the \: Sphere \: = \frac{3}{4} \times \frac{22}{7} \times ({r}^{3})=905.14}} \\ \\ \sf{\red{Volume \: of \: the \: Sphere \: = \frac{66}{28} \times ( {r}^{3} ) = 905.14 }} \\ \\ \\ \sf{\green{Volume \: of \: the \: Sphere = {r}^{3} {\implies \: 216}}} \\ \\ \\ \sf{\blue{Volume \: of \: the \: Sphere = (r) {\implies 6cm}}}$

$\sf{ \red{ \fbox{Hence, \: Required}}} \sf{ \green{ \fbox{ Radius \: of \: Sphere}}} \sf{ \blue{ \fbox{ = 6cm}}}$