Displacement in a cylinder by a sphere — find sphere radius: A cylindrical tub of radius 12 cm contains water to a depth of 20 cm. When a solid sphere is dropped in and completely submerged, the water level rises by 6.75 cm. Find the radius of the sphere.

Introduction / Context:
Submerging a solid displaces an equal volume of water. In a vertical cylinder, the rise in level converts readily to a displaced volume: V_displaced = πR_cyl^2 * Δh.

Given Data / Assumptions:

• Cylinder radius R_cyl = 12 cm
• Rise Δh = 6.75 cm
• Sphere volume V_sphere = (4/3)πr^3

Concept / Approach:
Set displaced volume equal to the sphere volume and solve for sphere radius r.

Step-by-Step Solution:
V_displaced = π*(12^2)*6.75 = π*144*6.75 = 972π cm3(4/3)πr^3 = 972π ⇒ r^3 = 972*(3/4) = 729r = 9 cm

Verification / Alternative check:
9^3 = 729 confirms r = 9 cm exactly.

Why Other Options Are Wrong:
6 cm and 8 cm under-displace; “None” is unnecessary with exact equality.

Common Pitfalls:
Forgetting π cancels; mis-multiplying 144 by 6.75.

Final Answer:
9 cm