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 radius $r'$ of the new sphere.

Because the twenty-seven smaller spheres are melted to form a single larger sphere, their total volume equals the volume of the new sphere. The combined volume is twenty-seven times four thirds times pi times r cubed, which simplifies to thirty-six times pi times r cubed. Equating this to the new sphere's volume formula, four thirds times pi times the new radius cubed, gives four times r cubed equals one-third times the new radius cubed. Multiplying both sides by three shows that the new radius cubed equals twenty-seven times r cubed, meaning the new radius r prime is 3r.

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