A right circular cylinder of radius 5 cm contains 10 solid spheres each of radius 5 cm, stacked so that the top sphere just touches the lid. Find the volume of the empty space in the cylinder.

Introduction / Context:We compare the volume of a cylinder to the total volume occupied by 10 identical spheres stacked exactly inside it. The empty space equals cylinder volume minus the spheres’ combined volume.

Given Data / Assumptions:

• Cylinder radius R = 5 cm.
• Each sphere radius r = 5 cm (fits snugly inside the cylinder).
• 10 spheres stacked vertically; the top touches the lid.

Concept / Approach:

• Cylinder height H equals 10 diameters = 10*(2r) = 100 cm.
• V_cyl = π*R^2*H = π*25*100 = 2500π cm^3.
• V_spheres_total = 10*(4/3)*π*r^3.
• Empty space = V_cyl − V_spheres_total.

Step-by-Step Solution:

Verification / Alternative check:Units are consistent (cm^3). Geometry fits: radius equality ensures no lateral gaps beyond those inherent to cylinder-sphere packing in a column; the height assumption is exact because spheres touch each other and the lids.

Why Other Options Are Wrong:

• 50000 π, 2500 π, 5000/3 π: Do not match the cylinder-minus-spheres calculation.
• None of these: Incorrect; (2500/3)π is exact.

Common Pitfalls:

• Using cylinder diameter instead of radius in volume.
• Forgetting to multiply the sphere volume by 10.

Final Answer:2500/3 π cm^3