![{\large{\textsf{\textbf{\underline{\underline{Given :}}}}}}](https://tex.z-dn.net/?f=%7B%5Clarge%7B%5Ctextsf%7B%5Ctextbf%7B%5Cunderline%7B%5Cunderline%7BGiven%20%3A%7D%7D%7D%7D%7D%7D)
★ Radius of first sphere
= 6cm.
★ Radius of second sphere
= 8cm.
★ Radius of third sphere
= 10cm.
![{\large{\textsf{\textbf{\underline{\underline{To \: Find :}}}}}}](https://tex.z-dn.net/?f=%20%7B%5Clarge%7B%5Ctextsf%7B%5Ctextbf%7B%5Cunderline%7B%5Cunderline%7BTo%20%5C%3A%20Find%20%3A%7D%7D%7D%7D%7D%7D)
★ The radius of the resulting sphere formed.
![{\large{\textsf{\textbf{\underline{\underline{Formula \: used :}}}}}}](https://tex.z-dn.net/?f=%20%7B%5Clarge%7B%5Ctextsf%7B%5Ctextbf%7B%5Cunderline%7B%5Cunderline%7BFormula%20%5C%3A%20used%20%3A%7D%7D%7D%7D%7D%7D)
![\star \: \tt Volume \: of \: sphere = {\underline{\boxed{\sf{\red{ \dfrac{ 4}{3}\pi {r}^{3} }}}}}](https://tex.z-dn.net/?f=%5Cstar%20%5C%3A%20%5Ctt%20Volume%20%5C%3A%20of%20%5C%3A%20sphere%20%3D%20%7B%5Cunderline%7B%5Cboxed%7B%5Csf%7B%5Cred%7B%20%5Cdfrac%7B%204%7D%7B3%7D%5Cpi%20%7Br%7D%5E%7B3%7D%20%20%7D%7D%7D%7D%7D)
![{\large{\textsf{\textbf{\underline{\underline{Concept :}}}}}}](https://tex.z-dn.net/?f=%20%7B%5Clarge%7B%5Ctextsf%7B%5Ctextbf%7B%5Cunderline%7B%5Cunderline%7BConcept%20%3A%7D%7D%7D%7D%7D%7D)
★ 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 :}}}}}}](https://tex.z-dn.net/?f=%20%7B%5Clarge%7B%5Ctextsf%7B%5Ctextbf%7B%5Cunderline%7B%5Cunderline%7BSolution%20%3A%7D%7D%7D%7D%7D%7D)
Let,
The radius of resulting sphere be ![R](https://tex.z-dn.net/?f=R)
<u>According</u><u> </u><u>to</u><u> </u><u>the</u><u> </u><u>question</u><u>,</u>
• Volume of first sphere + volume of second sphere + volume of third sphere = Volume of new sphere.
![\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}](https://tex.z-dn.net/?f=%20%5Clongrightarrow%20%5Csf%20%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%20%7B%28r_%7B1%7D%29%7D%5E%7B3%7D%20%20%2B%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%20%7B%28r_%7B2%7D%29%7D%5E%7B3%7D%20%2B%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%20%7B%28r_%7B3%7D%29%7D%5E%7B3%7D%20%3D%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%20%20%7B%28R%29%7D%5E%7B3%7D%20)
• here
☆
= 6cm
☆
= 8cm
☆
= 10cm
<u>Putting the values</u><u>,</u>
![\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}](https://tex.z-dn.net/?f=%20%5Clongrightarrow%20%5Csf%20%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%20%7B%286%29%7D%5E%7B3%7D%20%20%2B%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%20%7B%288%29%7D%5E%7B3%7D%20%2B%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%20%7B%2810%29%7D%5E%7B3%7D%20%3D%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%20%20%7B%28R%29%7D%5E%7B3%7D%20)
<u>Takin</u><u>g</u><u> </u>"
" <u>common,</u>
![\longrightarrow \sf \dfrac{4}{3} \pi \bigg[ {(6)}^{3} + {(8)}^{3} + {(10)}^{3} \bigg] = \dfrac{4}{3} \pi {(R)}^{3}](https://tex.z-dn.net/?f=%20%5Clongrightarrow%20%5Csf%20%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%20%5Cbigg%5B%20%7B%286%29%7D%5E%7B3%7D%20%20%2B%20%20%7B%288%29%7D%5E%7B3%7D%20%2B%20%20%7B%2810%29%7D%5E%7B3%7D%20%5Cbigg%5D%20%3D%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%20%20%7B%28R%29%7D%5E%7B3%7D%20)
![\longrightarrow \sf \cancel{ \dfrac{4}{3} \pi} \bigg[ {(6)}^{3} + {(8)}^{3} + {(10)}^{3} \bigg] = \cancel{ \dfrac{4}{3} \pi } {(R)}^{3}](https://tex.z-dn.net/?f=%5Clongrightarrow%20%5Csf%20%20%5Ccancel%7B%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%7D%20%5Cbigg%5B%20%7B%286%29%7D%5E%7B3%7D%20%20%2B%20%20%7B%288%29%7D%5E%7B3%7D%20%2B%20%20%7B%2810%29%7D%5E%7B3%7D%20%5Cbigg%5D%20%3D%20%5Ccancel%7B%20%5Cdfrac%7B4%7D%7B3%7D%20%5Cpi%20%7D%20%20%7B%28R%29%7D%5E%7B3%7D%20)
![\longrightarrow \sf \bigg[ {(6)}^{3} + {(8)}^{3} + {(10)}^{3} \bigg] = {(R)}^{3}](https://tex.z-dn.net/?f=%5Clongrightarrow%20%5Csf%20%20%20%5Cbigg%5B%20%7B%286%29%7D%5E%7B3%7D%20%20%2B%20%20%7B%288%29%7D%5E%7B3%7D%20%2B%20%20%7B%2810%29%7D%5E%7B3%7D%20%5Cbigg%5D%20%3D%20%20%7B%28R%29%7D%5E%7B3%7D%20)
![\longrightarrow \sf \bigg[ 216 +512 + 1000 \bigg] = {(R)}^{3}](https://tex.z-dn.net/?f=%5Clongrightarrow%20%5Csf%20%20%20%5Cbigg%5B%20216%20%20%2B512%20%2B%20%201000%20%5Cbigg%5D%20%3D%20%20%7B%28R%29%7D%5E%7B3%7D%20)
![\longrightarrow \sf 1728 = {(R)}^{3}](https://tex.z-dn.net/?f=%5Clongrightarrow%20%5Csf%20%20%201728%20%20%20%3D%20%20%7B%28R%29%7D%5E%7B3%7D%20)
![\longrightarrow \sf \sqrt[3]{1728} = R](https://tex.z-dn.net/?f=%5Clongrightarrow%20%5Csf%20%20%20%20%5Csqrt%5B3%5D%7B1728%7D%20%20%20%20%20%3D%20R)
![\longrightarrow \sf \sqrt[3]{ 12 \times 12 \times 12 } = R](https://tex.z-dn.net/?f=%5Clongrightarrow%20%5Csf%20%20%20%20%5Csqrt%5B3%5D%7B%2012%20%5Ctimes%2012%20%5Ctimes%2012%20%7D%20%20%20%20%20%3D%20R)
![\longrightarrow \sf \sqrt[3]{ {(12)}^{3} } = R](https://tex.z-dn.net/?f=%5Clongrightarrow%20%5Csf%20%20%20%20%5Csqrt%5B3%5D%7B%20%7B%2812%29%7D%5E%7B3%7D%20%7D%20%20%20%20%20%3D%20R)
![\longrightarrow \sf R = \red{12 \: cm}](https://tex.z-dn.net/?f=%5Clongrightarrow%20%5Csf%20%20%20%20%20R%20%3D%20%5Cred%7B12%20%5C%3A%20cm%7D%20%20%20)
Therefore,
<u>Radius of the resulting sphere is 12cm.</u>
![{\large{\textsf{\textbf{\underline{\underline{Note :}}}}}}](https://tex.z-dn.net/?f=%7B%5Clarge%7B%5Ctextsf%7B%5Ctextbf%7B%5Cunderline%7B%5Cunderline%7BNote%20%3A%7D%7D%7D%7D%7D%7D)
★ Figure in attachment.
![{\underline{\rule{290pt}{2pt}}}](https://tex.z-dn.net/?f=%7B%5Cunderline%7B%5Crule%7B290pt%7D%7B2pt%7D%7D%7D)