A solid metallic sphere of radius $8$ cm is melted and recast into spherical balls each of radius $2$ cm. The number of spherical balls, thus obtained, is

### Concept & Number of Recast Objects When a large object is melted to form smaller identical objects, the number of smaller objects is the total volume of the large object divided by the volume of one smaller object. $$n = \frac{V_{large}}{V_{small}}$$ ### Step-by-Step Solution * **Given:** Radius of large sphere $R = 8$ cm. Radius of small ball $r = 2$ cm. * Let the number of spherical balls be $n$. * $n \times \text{Volume of one small ball} = \text{Volume of large sphere}$. * $n \times \left(\frac{4}{3}\pi r^3\right) = \frac{4}{3}\pi R^3$ * The $\frac{4}{3}\pi$ terms cancel out on both sides: * $n \times (2^3) = (8^3)$ * $n = \frac{8^3}{2^3} = \left(\frac{8}{2}\right)^3$ * $n = (4)^3 = 64$. ### Exam Strategy & Shortcut When dividing volumes of similar 3D objects, ignore the constants ($\frac{4}{3}\pi$). The number of objects is simply the ratio of their defining linear dimensions cubed: $(R / r)^3 = (8 / 2)^3 = 4^3 = 64$. ### Common Pitfall Attempting to calculate the exact volume of the spheres by multiplying out $\frac{4}{3} \times \pi \times 512$ before dividing, which wastes time and increases the chance of calculation errors. ### Final Answer Therefore, the correct answer is **64**.