Multiple choice

A cylindrical box of radius 5 cm contains 10 solid spherical balls each of radius 5 cm. If the topmost ball touches the upper cover of the box, then the volume of the empty space in the box is :

  1. $\dfrac { 2500 }{ 3 } \pi { cm }^{ 3 }$
  2. $5000\pi { cm }^{ 3 }\quad $
  3. $2500\pi { cm }^{ 3 }\quad $
  4. $\dfrac { 5000 }{ 3 } \pi { cm }^{ 3 }$
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A Correct answer
Explanation

Cylinder radius 5, height 100 (10 balls of radius 5). Volume cylinder = pi * 5^2 * 100 = 2500pi. Volume 10 spheres = 10 * (4/3) * pi * 5^3 = 10 * (4/3) * 125 * pi = 5000pi/3. Empty space = 2500pi - 5000pi/3 = 2500pi/3.

AI explanation

Since the topmost ball touches the cover, the height of the cylindrical box equals the combined diameters of the 5 stacked balls, which is 5 * 10 = 50 cm. The volume of the cylinder is pi * R^2 * h = pi * 5^2 * 50 = 1250*pi cubic cm. The combined volume of the 10 spherical balls is 10 * 4/3 * pi * r^3 = 10 * 4/3 * pi * 5^3 = 5000/3*pi cubic cm. Subtracting the balls' volume from the cylinder's volume gives an empty space of 1250*pi - 5000/3*pi = (3750 - 5000)/3*pi, which equals 2500/3*pi cubic cm.

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