Multiple choice

$27$ metal balls each of radius r are melted together to form one big sphere of radius R. Then the ratio of surface area of the big sphere to that of a ball is __________.

  1. $9:1$
  2. $3:2$
  3. $\sqrt{27}:\sqrt{5}$
  4. $\sqrt{3}:1$
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A Correct answer
Explanation

Volume of 27 small spheres = Volume of 1 big sphere. 27 * (4/3 * pi * r^3) = 4/3 * pi * R^3. So R^3 = 27r^3, which means R = 3r. Surface area ratio = (4 * pi * R^2) / (4 * pi * r^2) = R^2 / r^2 = (3r)^2 / r^2 = 9/1.

AI explanation

Since volume is conserved, the volume of the large sphere equals 27 times the volume of a small ball, meaning (4/3) * pi * R^3 = 27 * (4/3) * pi * r^3, so R = 3r. The surface area of the big sphere is 4 * pi * R^2 and the surface area of a small ball is 4 * pi * r^2, making the ratio (3r)^2 : r^2, which is 9:1.

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