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Twenty-seven solid iron spheres, each of radius r and surface area S are melted to form a sphere with surface area S′. The radius r′ of the new sphere is ____.

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Detailed Solution

Assume that the radius, surface area and the volume of the new sphere is r′, S′, v′ respectively.
v =

\frac{4}{3} π r^{3}

- (1)
So, 27 spheres will have total volume of 27v=27×

\frac{4}{3} π r^{3}

=36

π r^{3}

.
v′= 27v =36

π r^{3}

- (2)
According to the radius of the new sphere r′, the volume of the bigger sphere: v′=

\frac{4}{3} π {(r^{'})}^{3}

From equation (1) and equation (2)

\frac{4}{3} π {(r^{'})}^{3} = 36

π r^{3}

⇒

{(r^{'})}^{3}

=27

r^{3}

(3 {r)}^{3}

⇒

{r'}^{'} = 3 r

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