The isotope 239 Pu 94 has a half life of 24,400 years. If a sample of nuclear waste originally contained 12kg of 239 Pu 94, how much time must pass before the amount falls to 3 kg?

Equation for half-lives:

Nt = No x (1/2) ^n

No = initial amount

Nt = final amount after t years

n = number of half lives = t / (single half-life)

t = years

Nt = 3/12 = 0.25

No = 12/12 = 1.00

n = t / (24400)

3/12 = (12/12) x (0.5) ^ (t/24400)

(0.25) = 1.00 x (0.5) ^ (t/24400)

0.25/1.00 = 0.5^ (t/24400)

ln (0.25) = ln (0.5^ (t/24400))

ln (0.25) = (t/24400) * ln (0.5)

ln (0.25) / ln (0.5) = (t/24400)

2 = t/24400

2*24400 = t

t = 48800 yrs

answer is t = 48,800 yrs