Multiple choice

In the science class, two students, Alex and Bella, are working on a project involving the study of spherical objects. Alex has a smaller hemispherical bowl (H1), and Bella has a larger hemispherical bowl (H2). If the ratio of the volumes of H1 to H2 is 8 : 27, what is the ratio of the curved surface areas of H1 to H2?

  1. 4 : 9

  2. 16 : 81

  3. 8 : 27

  4. 64 : 729

  5. 2 : 3
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A Correct answer
Explanation

The volume of a sphere is proportional to the cube of its radius (V proportional to r^3), so the ratio of radii is the cube root of the volume ratio. The cube root of 8:27 is 2:3. Surface area is proportional to the square of the radius (A proportional to r^2), so the ratio of surface areas is (2^2):(3^2), which is 4:9.

AI explanation

The volume of a hemisphere is proportional to the cube of its radius, so a volume ratio of 8 to 27 means the ratio of their radii is the cube root, which is 2 to 3. The curved surface area of a hemisphere is proportional to the square of its radius. Therefore, the ratio of their curved surface areas is 2 squared to 3 squared, resulting in a ratio of 4 to 9.

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