Question

A cone of height 24 and radius and base 6 cm is made up of modelling clay .A child reshapes it in the form of a sphere.Find the radius of sphere and surface area of sphere

Answer 1

$<b>$]
​$\it\huge\mathfrak\red{Given:-}$]

Height of cone=24

Radius=6cm

​$\it\huge\mathfrak\red{Solution:-}$]

Vol of cone=1/3πr^2h

=1/3π*6^2*24

=1/3π×36×24

= 288πcm^3

Let,

The vol of sphere = 4/3π^3

Vol of sphere = Vol of cone

4/3π^3 = 288π

4/3^3 = 288

r^3=288×3/4

r^3=72×3

r^3=8×9×3

r^3=(2×2×2×3×3×3)

r=(2×2×2×3×3×3)1/3

r=(2×2×2)1/3 × (3×3×3)1/3

r=(2^3)1/3 × (3^3)1/3

r=2×3

r=$\huge\boxed{\texttt{\fcolorbox{red}{white}{ 6 cm}}}$
$<Marquee>$]
Therefore , The radius if circle is 6cm

Answer 2

$\red{ \huge{ \underline{ \overline{ \mid{ \bold{ \purple{ \: \:SOLUTION\: \: \red{ \mid}}}}}}}}$

$\underline{ \underline{ \bold { \: \: GIVEN \: \: :}}} \to \\ \\ \bold{Height \: \: of \: \: the \: \: cone \: ,\: h \: = 24 \: cm} \\ \\ \bold{Radius \: \: of \: \: the \: \: cone \: ,\: r = 6 \: cm} \\ \\ \bold{ \therefore \: \: Volume \: \: of \: \: the \: \: cone = \frac{1}{3} \pi \: r {}^{2}h} \\ \\ \bold{ = [ \: \frac{1}{3} \: \pi \: \times (6) {}^{2} \times 2 4 \: ] \: \: cu. \: cm} \\ \\ \bold{ = ( \frac{1}{3 }\: \pi \: \times 36 \times 24) \: \: cu. \: cm } \\ \\ \bold{ = \frac{864 \: \pi}{3} \: \: cu. \: cm} \\ \\ \bold{ = 288 \: \pi \: \: cu. \: cm}$

$\bold{Suppose, } \\ \\ \bold{Radius \: \: of \: \: the \: \: sphere \: \: = r_1 } \\ \\ \bold{ \therefore \: \: Volume \: \: of \: \: the \: \: sphere = \frac{4}{3} \: \pi \: r_1 {}^{3} }$

$\underline{ \bold{ \: \: According \: \: \: To \: \: \: Question \: \: }} \\ \\ \bold{Volume \: \: of \: \: the \: \: Sphere = Volume \: \: of \: \: the \: \: Cone} \\ \\ \bold{ = > \: \: \: \: \: \: \frac{4}{3} \: \cancel\pi \: r_1 {}^{3} \: \: \: \: = \: \: \: \: 288 \: \cancel\pi } \\ \\ \bold{ = > \frac{4}{3} \: r_1{}^{3} = 288 } \\ \\ \bold{ = > \: \: \: r_1 {}^{3} \: \: \: = \: \: \: 288 \times \frac{3}{4} } \\ \\ \bold{ = > \: \: r_1 {}^{3 } \: \: \: = \: \: \: \frac{ \: \: 864 \: \: }{4} } \\ \\ \bold{ = > \: \: r_1 {}^{ 3 } \: \: \: = \: \: \: 216} \\ \\ \bold{ = > \: \: r_1 {}^{3} \: \: \: = \: \: \: (6) {}^{3} } \\ \\ \bold{ = > \: \: r_1 \: \: \: = \: \: \: 6} \\ \\ \\ \bold{ \therefore \: \: Radius \: \: of \: \: the \: \: sphere \: \: r_1 = 6 \: \: cm}$

$\bold{Now,} \\ \\ \bold{Surface \: \: area \: \: of \: \: the \: \: sphere = 4 \: \pi \: r_1 {}^{2} } \\ \\ \bold{ = [4 \times \frac{22}{7} \times (6){}^{2}] \: \: sq.cm } \\ \\ \bold{ = 452.57 \: \: \: sq .cm}$

$\red{ \huge{ \underline{ \overline{ \mid{ \bold{ \purple{ \: \: ANSWER \: \: \red{ \mid}}}}}}}}$

$\textsf{ \textbf{ \pink{(1) \: Radius \: \: of \: \: the \: \: sphere = 6 \: cm \: }}} \\ \\ \textsf{ \textbf{ \pink{(2) \: Surface \: \: area \: \: of \: \: the \: sphere = 452.57 \: \: sq.cm}}}$