The answer to this question can be explained using exponential decay, which is a mathematical model commonly used to describe the decay of radioactive materials.

Exponential Decay:
Exponential decay is a process in which the quantity of a substance decreases over time. It is characterized by a constant decay rate, meaning that the fraction of the substance that decays per unit of time remains the same. The equation for exponential decay is given by:

N(t) = N0 * e^(-λt)

Where:
- N(t) is the quantity of the substance at time t
- N0 is the initial quantity of the substance
- e is the base of the natural logarithm (approximately 2.71828)
- λ is the decay constant
- t is the time elapsed

Mean Life:
The mean life of a radioactive sample is the average time it takes for half of the sample to decay. In other words, it is the time at which the quantity of the sample is reduced to half its initial value. The mean life is related to the decay constant by the equation:

mean life = 1 / λ

In this case, the mean life is given as 100 years. Therefore, we can determine the decay constant:

100 years = 1 / λ
λ = 1 / 100 years

Percentage of the Sample Remaining:
To find the percentage of the sample remaining after 100 years, we substitute t = 100 years into the exponential decay equation:

N(t) = N0 * e^(-λt)

N(100) = N0 * e^(-λ * 100)

Since we are interested in the percentage remaining, we divide N(100) by N0 and multiply by 100:

Percentage remaining = (N(100) / N0) * 100

Substituting the values we have:

Percentage remaining = (N0 * e^(-λ * 100) / N0) * 100
Percentage remaining = e^(-λ * 100) * 100

Using the value of λ we found earlier:

Percentage remaining = e^(-1 * 100/100) * 100
Percentage remaining = e^(-1) * 100
Percentage remaining ≈ 0.36787944117 * 100
Percentage remaining ≈ 36.788% ≈ 37%

Therefore, after 100 years, approximately 37% of the radioactive sample remains active.