[tex]{\large{\textsf{\textbf{\underline{\underline{Given :}}}}}}[/tex]
★ Radius of first sphere [tex]\sf r_{1}[/tex] = 6cm.
★ Radius of second sphere [tex]\sf r_{2}[/tex] = 8cm.
★ Radius of third sphere [tex]\sf r_{3}[/tex] = 10cm.
[tex] {\large{\textsf{\textbf{\underline{\underline{To \: Find :}}}}}}[/tex]
★ The radius of the resulting sphere formed.
[tex] {\large{\textsf{\textbf{\underline{\underline{Formula \: used :}}}}}}[/tex]
[tex]\star \: \tt Volume \: of \: sphere = {\underline{\boxed{\sf{\red{ \dfrac{ 4}{3}\pi {r}^{3} }}}}}[/tex]
[tex] {\large{\textsf{\textbf{\underline{\underline{Concept :}}}}}}[/tex]
★ 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.
[tex] {\large{\textsf{\textbf{\underline{\underline{Solution :}}}}}}[/tex]
Let,
The radius of resulting sphere be [tex]R[/tex]
According to the question,
• Volume of first sphere + volume of second sphere + volume of third sphere = Volume of new sphere.
[tex] \longrightarrow \sf \dfrac{4}{3} \pi {(r_{1})}^{3} + \dfrac{4}{3} \pi {(r_{2})}^{3} + \dfrac{4}{3} \pi {(r_{3})}^{3} = \dfrac{4}{3} \pi {(R)}^{3} [/tex]
• here
☆[tex]\: \sf r_{1}[/tex] = 6cm
☆[tex]\: \sf r_{2}[/tex] = 8cm
☆[tex]\: \sf r_{3}[/tex] = 10cm
Putting the values,
[tex] \longrightarrow \sf \dfrac{4}{3} \pi {(6)}^{3} + \dfrac{4}{3} \pi {(8)}^{3} + \dfrac{4}{3} \pi {(10)}^{3} = \dfrac{4}{3} \pi {(R)}^{3} [/tex]
Taking " [tex]\dfrac{4}{3} \pi [/tex]" common,
[tex] \longrightarrow \sf \dfrac{4}{3} \pi \bigg[ {(6)}^{3} + {(8)}^{3} + {(10)}^{3} \bigg] = \dfrac{4}{3} \pi {(R)}^{3} [/tex]
[tex]\longrightarrow \sf \cancel{ \dfrac{4}{3} \pi} \bigg[ {(6)}^{3} + {(8)}^{3} + {(10)}^{3} \bigg] = \cancel{ \dfrac{4}{3} \pi } {(R)}^{3} [/tex]
[tex]\longrightarrow \sf \bigg[ {(6)}^{3} + {(8)}^{3} + {(10)}^{3} \bigg] = {(R)}^{3} [/tex]
[tex]\longrightarrow \sf \bigg[ 216 +512 + 1000 \bigg] = {(R)}^{3} [/tex]
[tex]\longrightarrow \sf 1728 = {(R)}^{3} [/tex]
[tex]\longrightarrow \sf \sqrt[3]{1728} = R[/tex]
[tex]\longrightarrow \sf \sqrt[3]{ 12 \times 12 \times 12 } = R[/tex]
[tex]\longrightarrow \sf \sqrt[3]{ {(12)}^{3} } = R[/tex]
[tex]\longrightarrow \sf R = \red{12 \: cm} [/tex]
Therefore,
Radius of the resulting sphere is 12cm.
[tex]{\large{\textsf{\textbf{\underline{\underline{Note :}}}}}}[/tex]
★ Figure in attachment.
[tex]{\underline{\rule{290pt}{2pt}}}[/tex]
