Question

The Surface area of a solid metallic sphere is 616 cm2. It is melted and recast into a cone
of height 28cm. Find the diameter of the base of cone so formed. [Use π = 22/7]

Answer 1

$\huge\purple{\underline{\underline{\pink{Ans}\red{wer:-}}}}$

$\sf{The \ diameter \ of \ base \ of \ cone \ is}$

$\sf{14 \ cm}$

$\sf\orange{\underline{\underline{Given:}}}$

$\sf{For \ sphere,}$

$\sf{\implies{Surface \ area=616 \ cm^{2}}}$

$\sf{For \ cone,}$

$\sf{\implies{Height (h)=28 \ cm}}$

$\sf\pink{\underline{\underline{To \ find:}}}$

$\sf{Diameter \ of \ base \ of \ cone.}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{Let \ radius \ of \ sphere \ be \ r \ and}$

$\sf{radius \ of \ cone \ be \ R.}$

$\sf{Surface \ area \ of \ sphere=4\pi\times \ r^{2}}$

$\sf{…formula}$

$\sf{616=4\times\frac{22}{7}\times \ r^{2}}$

$\sf{\therefore{r^{2}=\frac{616\times7}{4\times22}}}$

$\sf{r^{2}=49}$

$\sf{On \ taking \ square \ root \ of \ both \ sides}$

$\sf{r=7 \ cm}$

$\sf{Cone \ is \ formed \ by \ melting \ sphere.}$

$\sf{\therefore{Volume \ of \ sphere=Volume \ of \ cone}}$

$\sf{Volume \ of \ sphere=\frac{4}{3}\times \ \pi\times \ r^{3}}$

$\sf{...formula}$

$\sf{Volume \ of \ cone=\frac{1}{3}\times \ \pi\times \ R^{2}}$

$\sf{…formula}$

$\sf{\therefore{\frac{4}{3}\times\pi\times7^{3}=\frac{1}{3}\times\pi\times \ R^{2}\times28}}$

$\sf{R^{2}=\frac{4\times343}{28}}$

$\sf{\therefore{R^{2}=49}}$

$\sf{On \ taking \ square \ root \ of \ both \ sides}$

$\sf{R=7 \ cm}$

$\sf{Diameter=2R=2(7)}$

$\sf{\therefore{Diameter=14 \ cm}}$

$\sf\purple{\tt{\therefore{The \ diameter \ of \ base \ of \ cone \ is}}}$

$\sf\purple{\tt{14 \ cm}}$

Answer 2

$\large\bf\underline \orange{Given:-}$

• Surface area of sphere = 616cm²
• Height of cone = 28cm

$\large\bf\underline \orange{To \: find:-}$

• diameter of base of cone.

$\huge\bf\underline \green{Solution:-}$

we know that,

$\boxed{\boxed{ \pink{ \bold {Surface \: Area \: of \: Sphere} = \green{4 \pi \: r^{2}}}}} \\\\ : \: \mapsto \rm \:616 = 4 \times \frac{22}{7} \times {r}^{2} \\ \\ : \: \mapsto \rm \frac{ \cancel{616}}{4} \times \frac{7}{ \cancel{22}} = {r}^{2} \\ \\ : \: \mapsto \rm \: \frac{308}{4} \times \frac{7}{11} = {r}^{2} \\ \\ : \: \mapsto \rm \: \cancel\frac{2156}{44} = {r}^{2} \\ \\ : \: \mapsto \rm \:49 = {r}^{2} \\ \\ : \: \mapsto \rm \:r \: = \sqrt{49} \\ \\ : \: \mapsto \bf \blue{\: r = 7cm }$

• Volume of sphere = Volume of cone

$\boxed{\boxed{ \pink{ \bold {Volume \: of \: Cone } = \green{ \frac{1}{3} \bf\pi \: r^{2}h}}}}$

$:\:\mapsto\:\rm\:volume \: of \: cone \: = \frac{1}{3} \pi \: {r}^{2} \times 28$

$\boxed{\boxed{ \pink{ \bold {Volume \: of \: Sphere } = \green{ \frac{4}{3} \bf\pi \: r^{3}}}}}$

$:\:\mapsto\:\rm\:volume \: of \: cone \: = \frac{4}{3} \pi \: {7}^{3}$

As we know that ,

• volume of sphere = volume of cone.

$:\:\mapsto\:\rm \: \frac{4}{3} \pi \: \times {7}^{3} = \frac{1}{3} \pi \: {r}^{2} \times 28 \\\\:\:\mapsto\:\rm \: \frac{4 \cancel\pi}{3} \times \frac{3}{ \cancel \pi} = {r}^{2} \times \dfrac{28} { {7}^{3} } \\ \\:\:\mapsto\:\rm \:4 \times \frac{ {7}^{3} }{28} = {r}^{2} \\ \\:\:\mapsto\:\rm \: {7}^{2} = {r}^{2} \\ \\:\:\mapsto\:\bf \pink{r = 7} \:$

Diameter of cone = 2r

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀= 2× 7

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀=14cm⠀