We can use the half-life formula to solve this problem:
N = N0 * (1/2)^(t/T)
where:
N = final number of nuclei (250)
N0 = initial number of nuclei (4,000)
t = time elapsed
T = half-life (10 minutes)
Substituting the given values into the formula:
250 = 4,000 * (1/2)^(t/10)
Dividing both sides by 4,000:
1/16 = (1/2)^(t/10)
Taking the logarithm of both sides (base 2):
log(1/16) = log[(1/2)^(t/10)]
-4 = (t/10) * log(2)
Solving for t:
t = -4 / log(2) * 10
t ≈ 13.86 minutes
Therefore, you were gone from the laboratory for approximately 13.86 minutes.