A conical cavity of maximum volume is carved out from a wooden solid hemisphere of radius 10 cm. Curved surface area of the cavity carved out is (use $\pi = 3.14$)
Generated by claude-sonnet-4-6 · 2026-06-15 10:36 · grounding rag
Model Answer
For maximum volume, the cone carved from a solid hemisphere has radius = 10 cm and height = 10 cm.
Slant height $l = \sqrt{r^2 + h^2} = \sqrt{10^2 + 10^2} = 10\sqrt{2}$ cm
CSA of conical cavity $= \pi r l = 3.14 \times 10 \times 10\sqrt{2} = 314\sqrt{2}$ cm²
(A) $314\sqrt{2}$ cm²
Source: Chapter 12, Section 12.2
Explanation
- The largest cone that fits inside a hemisphere of radius $R$ has base radius $r = R$ and height $h = R$ (the apex touches the curved surface and the base coincides with the flat face).
- Only the curved (lateral) surface area of the cone is asked — use $\pi r l$, not total surface area.
- Slant height $l = \sqrt{R^2 + R^2} = R\sqrt{2}$, giving CSA $= \pi R \cdot R\sqrt{2} = \pi R^2\sqrt{2} = 3.14 \times 100 \times \sqrt{2} = 314\sqrt{2}$ cm².