Multiple choice

The diameters of external and internal surfaces of a hollow spherical shell are 10 cm and 6 cm respectively. If it is melted and recast into a solid cylinder of length of $2\dfrac {2}{3}cm$, find the diameter of the cylinder in cm

  1. $14$
  2. $16$
  3. $18$
  4. $20$
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A Correct answer
Explanation

Volume of hollow sphere = (4/3) * pi * (R^3 - r^3) = (4/3) * pi * (5^3 - 3^3) = (4/3) * pi * (125 - 27) = (4/3) * pi * 98 = 392/3 * pi. Cylinder volume = pi * r_c^2 * h = pi * r_c^2 * (8/3). Setting equal: 392/3 * pi = 8/3 * pi * r_c^2, so r_c^2 = 392/8 = 49. Radius = 7, diameter = 14.

AI explanation

Using the volume conservation method, the volume of the hollow spherical shell equals the volume of the new solid cylinder. The shell volume is (4/3) * pi * (5^3 - 3^3) = (4/3) * pi * 98 = (392/3) * pi. For a cylinder of length (8/3) cm, this volume equals pi * R^2 * (8/3), meaning R^2 = 49 and the radius R is 7 cm. The required diameter is twice the radius, which is 14 cm.

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