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.

Kindly help!​

Answer 1

`{\large{\textsf{\textbf{\underline{\underline{Given :}}}}}}`

★ Radius of first sphere
`\sf r_(1)` = 6cm.

★ Radius of second sphere
`\sf r_(2)` = 8cm.

★ Radius of third sphere
`\sf r_(3)` = 10cm.

`{\large{\textsf{\textbf{\underline{\underline{To \: Find :}}}}}}`

★ The radius of the resulting sphere formed.

`{\large{\textsf{\textbf{\underline{\underline{Formula \: used :}}}}}}`

`\star \: \tt Volume \: of \: sphere = {\underline{\boxed{\sf{\red{ ( 4)/(3)\pi {r}^(3) }}}}}`

`{\large{\textsf{\textbf{\underline{\underline{Concept :}}}}}}`

★ As, three spheres are melted to from one new sphere. Therefore, volume of three old sphere is equal to volume of new sphere.

i.e, Volume of first sphere + volume of second sphere + volume of third sphere = Volume of new sphere.

`{\large{\textsf{\textbf{\underline{\underline{Solution :}}}}}}`

Let,

The radius of resulting sphere be
`R`

According to the question,

• Volume of first sphere + volume of second sphere + volume of third sphere = Volume of new sphere.

`\longrightarrow \sf (4)/(3) \pi {(r_(1))}^(3) + (4)/(3) \pi {(r_(2))}^(3) + (4)/(3) \pi {(r_(3))}^(3) = (4)/(3) \pi {(R)}^(3)`

• here

☆
`\: \sf r_(1)` = 6cm

☆
`\: \sf r_(2)` = 8cm

☆
`\: \sf r_(3)` = 10cm

Putting the values,

`\longrightarrow \sf (4)/(3) \pi {(6)}^(3) + (4)/(3) \pi {(8)}^(3) + (4)/(3) \pi {(10)}^(3) = (4)/(3) \pi {(R)}^(3)`

Taking "
`(4)/(3) \pi`" common,

`\longrightarrow \sf (4)/(3) \pi \bigg[ {(6)}^(3) + {(8)}^(3) + {(10)}^(3) \bigg] = (4)/(3) \pi {(R)}^(3)`

`\longrightarrow \sf \cancel{ (4)/(3) \pi} \bigg[ {(6)}^(3) + {(8)}^(3) + {(10)}^(3) \bigg] = \cancel{ (4)/(3) \pi } {(R)}^(3)`

`\longrightarrow \sf \bigg[ {(6)}^(3) + {(8)}^(3) + {(10)}^(3) \bigg] = {(R)}^(3)`

`\longrightarrow \sf \bigg[ 216 +512 + 1000 \bigg] = {(R)}^(3)`

`\longrightarrow \sf 1728 = {(R)}^(3)`

`\longrightarrow \sf \sqrt[3]{1728} = R`

`\longrightarrow \sf \sqrt[3]{ 12 * 12 * 12 } = R`

`\longrightarrow \sf \sqrt[3]{ {(12)}^(3) } = R`

`\longrightarrow \sf R = \red{12 \: cm}`

Therefore,

Radius of the resulting sphere is 12cm.

`{\large{\textsf{\textbf{\underline{\underline{Note :}}}}}}`

★ Figure in attachment.

`{\underline{\rule{290pt}{2pt}}}`