Q1. [1] § 12.4 Summary
A cone of maximum size is carved out from a solid cube of edge length $l$. The volume of the cone is :
- A $\frac{\pi l^3}{12}$
- B $\frac{\pi l^3}{3}$
- C $l^3\left(1 - \frac{\pi}{3}\right)$
- D $\frac{\pi l^3}{8}$
Previously asked in CBSE board exam
2026 30/3/1 Q7
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.
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