Multiple choice

A hollow spherical ball whose inner radius is $4\ cm$ is full of water. Half of the water is transferred to a conical cup and it completely filled the cup. If the height of the cup is $2\ cm$, then the radius of the base of cone, in $cm$ is:

  1. $4cm$
  2. $8\pi cm$
  3. $8cm$
  4. $16cm$
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C Correct answer
Explanation

The volume of the hollow sphere is (4/3) * pi * r^3. With r=4, volume = (4/3) * pi * 64 = 256pi/3. Half of this is 128pi/3. The volume of the cone is (1/3) * pi * R^2 * h. Setting (1/3) * pi * R^2 * 2 = 128pi/3 leads to 2 * R^2 = 128, so R^2 = 64 and R = 8.

AI explanation

The volume of water in the conical cup equals half the volume of the spherical water, giving the equation one third multiplied by pi multiplied by the radius squared multiplied by 2 equals half of four thirds multiplied by pi multiplied by 4 cubed. Simplifying the right side results in 128 multiplied by pi divided by 3, so equating both sides and cancelling pi gives two thirds multiplied by the radius squared equals 128 divided by 3. Solving this yields a radius squared of 64, making the radius 8 centimeters.

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