At a certain instant, a piece of radioactive material contains 10¹² atoms. The half-life of the material is 30 days. How long will elapse before 10⁴…

Decay follows N = N₀ × (½)^(t/T), where T = 30 days.

N ÷ N₀ = 10⁴ ÷ 10¹² = 10⁻⁸. So 2^(t/30) = 10⁸.

Number of half-lives = log 10⁸ ÷ log 2 = 8 ÷ 0.301 ≈ 26.6.

Time = 26.6 × 30 ≈ 800 days, so the answer is 801 days (small differences come from rounding log 2).