Question

Metallic spheres of radii 6 cm, 8 cm and 10 cm respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere.​

Answer 1

QuestioN

• Metallic spheres of radii 6 cm, 8 cm and 10 cm respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere.

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$\pmb{\mathcal{Given}}\begin{cases}\sf{\:\succ\:Radius,\:(r_{1})_{(1^{st}\:sphere)}\:=\:6\:cm}&\\ \frak {\:\:Radius,\:(r_{2})_{(2^{nd}\:sphere)}\:=\:8\:cm}&\\ \sf{\:\succ\:Radius,\:(r_{3})_{(3^{rd}\:sphere)}\:=\:10\:cm}\end{cases}$

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$\normalsize{\underline{\pmb{\mathcal{To\:find,}}}}$

• Radius of the sphere so formed will be ?

$\:\:\:\:\:\:\:\:───────────────────$

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${\bold \bullet\: { \underline{\small{Let\:the\: radius\:of\:the\: resulting\: sphere\:be\:r }}}}$

$\sf {\large\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \curvearrowright}$

$\big| \sf The \:object \:formed\: by\: recasting\: these \: \:&\\ \sf \:\:spheres\:will\: be\: same\: in \:volume \:as\: the\: sum\: &\\ \sf of\: the\: volumes\: of\:these\:spheres.\big|$

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$\normalsize{\underline{\pmb{\mathcal{Now, }}}}$

$\:\:\:\:\dag \small{\underline{\boxed{\sf{Vol.\:of\:3\: spheres =\frak{\red{Vol.\:of\: resulting\:sphere}}}}}}$

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$\sf \:\:\:\:\:\:\:\:\:\:\nrightarrow\: \bf\dfrac{4}{3} \pi \bigg[{r_1}^3+{r_2}^3+{r_3}^3 \bigg]\:=\:\bf\dfrac{4}{3} \pi r^{3}$

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$\sf \:\:\:\:\:\:\:\:\:\:\twoheadrightarrow\:\cancel{\bf\dfrac{4}{3} \pi } \bigg[ 6^3\:+\:8^3\:+\:10^3 \bigg]\:=\:\cancel{\bf\dfrac{4}{3} \pi} r^{3}$

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$\sf \:\:\:\: \looparrowright\:\: r^3\:=\:216\:+\:512\:+\:1000\:=\:1728$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\small{\underline{\pmb{\mathcal{\green{r\:=\:12\:cm }}}}}$

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Therefore, the radius of the sphere so formed will be 12 cm.

Answer 2

$\huge \fbox \orange{Answer✪}$

$\textbf{Given: } \: R_{1} = 6 \: cm ,R_{2} = 8 \: cm,R_{3} = 10 \: cm$

$\textbf{To Find: } \text{Radius of Resulting sphere}$

$\sf \colorbox{skyblue} {Solution: }$

$\text{Volume \: 2} = \frac{4}{3}\pi {R_{1}}^{3}$

$⇒ \frac{4}{3} \times \frac{22}{7} \times 6 \times 6 \times 6 = 904.78 \: {cm}^{3}$

$\text{Volume \: 2} = \frac{4}{3}\pi {R_{2}}^{3}$

$⇒ \frac{4}{3} \times \frac{22}{7} \times {8}^{3} = 2144.66 \: {cm}^{3}$

$\text{Volume \: 3} = \frac{4}{3}\pi {R_{3}}^{3}$

$⇒ \frac{4}{3} \times \frac{22}{7} \times {10}^{3} = 4188.88.80 \: {cm}^{3}$

$\textbf{Total Volume:}$

$⇒904.78 + 2144.66 + 4188.80$

$= 7238.23 \: {cm}^{3} →(1)$

$∴\text{Volume of resulting sphere} = \frac{4}{3}\pi {r}^{3}$

$⇒7238.23 = \frac{4}{3} \times \frac{22}{7} \times {r}^{3}$

$⇒ {r}^{3} = \frac{7238.23 \times 3 \times 4}{4 \times 22} = 1724.30$

$\fbox{Hence Redius is 12 cm}$

$\\ \\ \\ \\ \sf \colorbox{lightgreen} {\red★ANSWER ᵇʸɴᴀᴡᴀʙﷻ}$