Discussion
A solid sphere of radius 10 cm is melted and recast into small solid spheres of radius 0.5 cm. How many small spheres are formed?
Answer: A
8000 small spheres are formed. Melting and recasting keeps the total volume the same, so the number of small spheres equals the ratio of the volumes. Since the volume of a sphere is proportional to the cube of its radius, the common factor \(\frac{4}{3}\pi\) cancels.
\[ n = \left(\frac{10}{0.5}\right)^{3} = 20^{3} = 8000 \]The radius ratio is 20, so the count is 20 cubed. Using 20 squared would give 400, and none of 800, 1600 or 16000 is 20 cubed.
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