The volume of the greatest sphere that can be cut off from a cylindrical log of wood of base radius 1 cm and height 5 cm is

### Concept & Inscribed Geometries To cut the "greatest sphere" from a cylinder, the sphere's diameter cannot exceed the smallest dimension of the cylinder. It is constrained by either the cylinder's diameter or its height. $$Volume_{sphere} = \frac{4}{3}\pi r^3$$ ### Step-by-Step Solution 1. Identify the cylinder's constraints: Base radius = 1 cm (Diameter = 2 cm), Height = 5 cm. 2. The greatest sphere must fit entirely inside this cylinder. 3. If the sphere's diameter were equal to the cylinder's height (5 cm), it would not fit inside the cylinder's 2 cm diameter width. 4. Therefore, the maximum diameter of the sphere is limited by the cylinder's diameter, which is 2 cm. 5. The radius of this maximal sphere is 1 cm. 6. Calculate the volume of this sphere: $V = \frac{4}{3}\pi (1)^3 = \frac{4}{3}\pi$. ### Exam Strategy & Shortcut Always look for the smallest limiting dimension when fitting shapes inside one another. The limiting factor here is the diameter of 2 cm. The height of 5 cm is extra space that cannot be utilized by a perfect sphere. ### Common Pitfall Assuming the maximum sphere volume involves averaging the dimensions or attempting to equate volumes, instead of recognizing it as a geometric constraint problem limited by the smallest internal width. ### Final Answer Therefore, the correct answer is **$\frac{4}{3}\pi$**.