A cylinder whose height is eight times its radius is melted and cast into spherical balls each of half the radius of the cylinder. Find the number of spherical balls.
48
Let the radius of the cylinder be \(r\). Then its height is \(h = 8r\), and the radius of each spherical ball is \(\frac{r}{2}\).
Volume of the cylinder \(= \pi r^2 h = \pi r^2 (8r) = 8\pi r^3\).
Volume of one spherical ball \(= \frac{4}{3}\pi \left(\frac{r}{2}\right)^3 = \frac{4}{3}\pi \cdot \frac{r^3}{8} = \frac{\pi r^3}{6}\).
Since melting conserves volume, the number of balls \(= \dfrac{\text{Volume of cylinder}}{\text{Volume of one ball}} = \dfrac{8\pi r^3}{\frac{\pi r^3}{6}}\).
This gives \(= 8\pi r^3 \times \dfrac{6}{\pi r^3} = 48\).
Hence, the number of spherical balls is 48.
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