A solid iron ball is melted and 64 smaller solid balls of equal size are made using the entire volume of iron. What is the ratio of the surface area of the larger ball to the sum of the surface areas of all the smaller balls?

Explanation

Since volume is conserved, \frac{4}{3}\pi R^3 = 64 \times (\frac{4}{3}\pi r^3) \implies R^3 = 64r^3 \implies R = 4r. Surface area of the large ball = 4\pi R^2 = 4\pi(4r)^2 = 64\pi r^2. The sum of the surface areas of 64 smaller balls = 64 \times 4\pi r^2 = 256\pi r^2. The ratio is \frac{64\pi r^2}{256\pi r^2} = \frac{1}{4} = 0.25.

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