Multiple choice

The surface areas of a solid sphere and a solid hemisphere are equal to S if their volumes are $\displaystyle V_{1}$ and $\displaystyle V_{2}$ respectively then $\displaystyle \frac{V_{1}}{V_{2}}$.

  1. $\displaystyle \sqrt{3}:2$
  2. $\displaystyle 3\sqrt{3}:8$
  3. $3 : 4$
  4. $\displaystyle 3\sqrt{3}:4$
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D Correct answer
Explanation

For a sphere of radius r1, S = 4 * pi * r1^2. For a hemisphere of radius r2, S = 3 * pi * r2^2. Equating them, 4 * pi * r1^2 = 3 * pi * r2^2, so r1^2 / r2^2 = 3 / 4, or r1 / r2 = sqrt(3) / 2. The volume ratio V1 / V2 = ((4/3) * pi * r1^3) / ((2/3) * pi * r2^3) = 2 * (r1 / r2)^3 = 2 * (sqrt(3)/2)^3 = 2 * (3 * sqrt(3) / 8) = 3 * sqrt(3) / 4.

AI explanation

Let the sphere radius be R and the hemisphere radius be r, so 4 * pi * R^2 = 3 * pi * r^2 yields R = r * sqrt(3) / 2. Using the volume formulas, V1 is (4/3) * pi * R^3 and V2 is (2/3) * pi * r^3. Substituting R gives V1 / V2 = 2 * (r * sqrt(3) / 2)^3 / r^3, which simplifies to 3 * sqrt(3) / 4.

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