find the percentage of reduction in surface area
Q.6. Three solid metal spheres of radii 3 cm, 4 cm and 5 cm respectively are melted together. The obtained metal is recast into a single solid sphere. The percentage reduction in the surface area of the solid sphere is
A. 24 %
B. 26 %
C. 28 %
D. 30 %

Dear Student,

Please find below the solution to the asked query:

We know :  Surface area of sphere  =  4 $\mathrm{\pi }$ r²

So,

Surface area of sphere with radius 3 cm = 4 $\mathrm{\pi }$ $\times$ ( 3 )² =  36 $\mathrm{\pi }$ cm²   ,

Surface area of sphere with radius 4 cm = 4 $\mathrm{\pi }$ $\times$ ( 4 )² =  64 $\mathrm{\pi }$ cm²  

And

Surface area of sphere with radius 5 cm = 4 $\mathrm{\pi }$ $\times$ ( 5 )² =  100 $\mathrm{\pi }$ cm²  

Then,

Total surface area of all three spheres = 36 $\mathrm{\pi }$ + 64 $\mathrm{\pi }$+ 100 $\mathrm{\pi }$  =  200 $\mathrm{\pi }$ cm² 

We know volume of sphere = $\frac{4}{3}$$\times$ $\mathrm{\pi }$r³  

As given : Three solid metal spheres of radii 3 cm, 4 cm and 5 cm respectively are melted together. The obtained metal is recast into a single solid sphere. So

Volume of all three sphere =  Volume of new sphere

We assume radius of new sphere =  r  , So

$\frac{4}{3}$$\times$ $\mathrm{\pi }$ $\times$ ( 3 )³   + $\frac{4}{3}$$\times$ $\mathrm{\pi }$ $\times$ ( 4 )³   + $\frac{4}{3}$$\times$ $\mathrm{\pi }$ $\times$ ( 5 )³   = $\frac{4}{3}$$\times$ $\mathrm{\pi }$r³  ,

$\frac{4}{3}$$\times$ $\mathrm{\pi }$ [ ( 3 )³   +  ( 4 )³   + ( 5 )³  ] = $\frac{4}{3}$$\times$ $\mathrm{\pi }$r³  ,

[ 27 +  64  + 125 ] = r³  ,

[216 ] = r³  ,

r = 6  cm

Then,

Surface area of new sphere = 4 $\mathrm{\pi }$ $\times$ ( 6 )² =  144 $\mathrm{\pi }$ cm²  

Change in surface area =  200 $\mathrm{\pi }$ -  144 $\mathrm{\pi }$ =  56$\mathrm{\pi }$ cm²

Thus ,

Percentage reduction in the surface area of the solid sphere = $\frac{56 \mathrm{\pi }}{200 \mathrm{\pi }} \times 100 = \frac{{\displaystyle 56}}{{\displaystyle 200}} \times 100= \frac{{\displaystyle 56}}{{\displaystyle 2}} =\mathbf{ }\mathbf{28}\mathbf{ }\mathbf{\%}$
Therefore,

Option ( C )                                                          ( Ans )

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