Multiple choice

The ratio of whole surface area of a certain cube is equal to the area of the curved surface area of a certain sphere. Then ratio of their volumes is

  1. $\pi:6$
  2. $\sqrt{\pi}:\sqrt{6}$
  3. $\sqrt{6}:\sqrt{\pi}$
  4. $6:\pi$
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B Correct answer
Explanation

Surface area of cube = 6a^2. Surface area of sphere = 4*pi*r^2. 6a^2 = 4*pi*r^2 => a^2/r^2 = 4*pi/6 = 2*pi/3. Ratio of volumes = a^3 / ((4/3)*pi*r^3) = (a/r)^3 * (3/(4*pi)). (a/r) = sqrt(2*pi/3). Ratio = (2*pi/3)^(3/2) * 3/(4*pi) = (2*pi/3) * sqrt(2*pi/3) * 3/(4*pi) = 0.5 * sqrt(2*pi/3) = sqrt( (1/4) * (2*pi/3) ) = sqrt(pi/6).

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