A cone of maximum size is carved out from a solid cube of edge length $l$. The volume of the cone is :
Generated by claude-sonnet-4-6 · 2026-06-15 10:34 · grounding rag
Model Answer
Option A: $\dfrac{\pi l^3}{12}$
The maximum cone has base radius $r = \dfrac{l}{2}$ (inscribed circle of top face) and height $h = l$.
$$V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi \left(\frac{l}{2}\right)^2 \times l = \frac{\pi l^3}{12}$$
Explanation
- The largest cone carved from a cube of edge $l$ has its circular base inscribed in one face of the cube, giving radius $= l/2$, and height $= l$ (the edge length).
- Substitute into $V = \frac{1}{3}\pi r^2 h$ to get $\frac{\pi l^3}{12}$.
- A common mistake is taking $r = l$; remember the radius is half the edge since the circle is inscribed in the square face.