(a) A charcoal sample from an ancient hearth shows ¹⁴C activity 0.25 times that of fresh wood. Estimate its age. (T₁/₂ = 5730 yr.) (b) Why can't carbon dating date a 100,000-year-old fossil? Solution: (a) Ratio 0.25 = (1/2)² → 2 half-lives elapsed: t = 2 × 5730 = 11,460 years. (b) 100,000 years ≈ 17.5 half-lives → remaining fraction (1/2)^17.5 ≈ 5×10⁻⁶ — far too little ¹⁴C left to measure reliably; also the method only works on once-living material, not rocks.