Exam-Style Challenging Problems
Here we tackle high-frequency exam questions that combine sphere volume with density, liquid displacement, and multi-shape transformations.
Problem 1: Water Displacement in a Cylinder
A solid iron ball of radius r = 3 cm is dropped into a cylinder of radius R = 6 cm containing water. By how much does the water level rise?
1. Volume of iron sphere: V_sphere = 4/3 × π × 3³ = 36π cm³
2. The displaced water forms a cylinder of radius R = 6 and height h: V_water = π × R² × h = π × 6² × h = 36π h
3. Equating both: 36π h = 36π ⇒ h = 1 cm
Answer: The water level rises by exactly 1 centimeter.