Math, asked by coseti1787, 10 months ago

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]

Answers

Answered by Anonymous
4

\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}}

Answered by Anonymous
10

 \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

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