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Probability

Question
CBSEENMA9003709
Wired Faculty App

Twenty seven solid iron spheres, each of radius r and surface area S are melted to form a sphere with surface area S'. Find the

(i)     radius r´ of the new sphere,
(ii)    ratio of S and S’. 

Solution
(i) Volume of a solid iron sphere equals 4 over 3 πr cubed
∴ Volume of 27 solid iron spheres = 27 open parentheses 4 over 3 πr cubed close parentheses equals 36 πr cubed
∴ Volume of the new sphere = 36πr3 Let the radius of the new sphere be r’. Then,
Volume of the new sphereequals 4 over 3 πr apostrophe cubed
According to the question,
                   
                4 over 3 πr apostrophe cubed equals 36 πr cubed
rightwards double arrow space space space space space space space space space space space space space space space space space straight r apostrophe cubed equals fraction numerator left parenthesis 36 πr cubed right parenthesis cubed over denominator 4 straight pi end fraction rightwards double arrow space space space space space space space space space space space space space space space space space straight r apostrophe cubed equals 27 straight r cubed rightwards double arrow space space space space space space space space space space space space space space space space space straight r apostrophe equals left parenthesis 27 straight r cubed right parenthesis to the power of bevelled 1 third end exponent rightwards double arrow space space space space space space space space space space space space space space space space space straight r apostrophe equals left parenthesis 3 cross times 3 cross times 3 straight r cubed right parenthesis to the power of begin inline style bevelled 1 third end style end exponent rightwards double arrow space space space space space space space space space space space space space space space space space space straight r apostrophe equals 3 straight r
Hence, the radius r’ of the new-sphere is 3r.
(ii) S = 4πr2 S’ = 4π(3r)2
therefore space space space fraction numerator straight S over denominator straight S apostrophe end fraction equals fraction numerator 4 πr squared over denominator 4 straight pi left parenthesis 3 straight r right parenthesis squared end fraction equals 1 over 9 equals 1 space colon space 9
Hence, the ratio of S and S’ is 1 : 9.

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