(a) To determine the half-life of the substance, we can use the formula:
N(t) = N0 * (1/2)^(t/T)
where N(t) is the amount of substance remaining after time t, N0 is the initial amount of substance, T is the half-life, and (1/2) is the decay constant.
We know that after one year, the substance has decayed to 95.5% of its original amount, so:
N(t) = 0.955 * N0
Substituting this into the formula, we get:
0.955 * N0 = N0 * (1/2)^(1/T)
Simplifying and solving for T, we get:
T = ln(2) / ln(1/2) ≈ 0.693
Therefore, the half-life of the substance is approximately 0.693 years.
(b) To determine how long it would take the sample to decay to 55% of its original amount, we can again use the formula:
N(t) = N0 * (1/2)^(t/T)
We know that after this amount of time, the substance will have decayed to 55% of its original amount, so:
N(t) = 0.55 * N0
Substituting this into the formula, we get:
0.55 * N0 = N0 * (1/2)^(t/T)
Simplifying and solving for t, we get:
t = T * ln(0.55) / ln(1/2) ≈ 1.48 years
Therefore, it would take approximately 1.48 years for the sample to decay to 55% of its original amount.