Multiple choice

A right circular cylinder is partially filled with water. Two iron spherical balls are completely immersed in the water so that the height of the water in the cylinder rises by 4 cm. If the radius of one ball is half of the other and the diameter of the cylinder is 18 cm, then the radii of the spherical balls are

  1. 6 cm and 12 cm

  2. 4 cm and 8 cm

  3. 3 cm and 6 cm

  4. 2 cm and 4 cm
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C Correct answer
Explanation

Volume of water rise = pi * r^2 * h = pi * 9^2 * 4 = 324pi. Volume of two spheres = 4/3 * pi * (r^3 + (2r)^3) = 4/3 * pi * 9r^3 = 12pi * r^3. 12pi * r^3 = 324pi => r^3 = 27 => r = 3. Radii are 3 cm and 6 cm.

AI explanation

The volume of the displaced water equals the volume of the two iron balls. Using the cylinder volume formula, the displaced water volume is pi * 9^2 * 4 = 324*pi. Using the sphere volume formula, the balls' total volume is (4/3)pi*r^3 + (4/3)*pi(2r)^3 = (4/3)pi(9r^3) = 12*pi*r^3. Setting the volumes equal gives 12*pi*r^3 = 324*pi, so r^3 = 27 and the smaller radius is r = 3 cm. The larger radius is 2r = 6 cm, making the radii 3 cm and 6 cm.

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