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Half-life is a fundamental concept in understanding exponential decay, especially in the context of radioactive materials. Simply put, half-life is the time required for a quantity to reduce to half its initial amount. In real-world terms, it's how long it takes for half of a radioactive substance to undergo decay and turn into something else.

For strontium-90, which is a byproduct of nuclear reactions, the half-life can be calculated using the formula for exponential decay. The formula is:

• \[ P(t) = P_0 (0.975)^t \]

This represents the percentage of strontium-90 remaining after time \( t \), given a decay rate of 2.5\% per year.

To find the half-life, you set \( P(t) = 50 \) in this formula, because you are looking for the time when only 50\% of the original amount is left. By solving the equation \[ 0.5 = (0.975)^t \] using logarithms, you find that the half-life of strontium-90 is approximately 27.72 years. This means after this time frame, half of the initial strontium-90 has decayed.

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. In the case of strontium-90, as it decays, it transforms into another element. This process follows an exponential decay model.This model is represented by the equation \[ P(t) = P_0 e^{-kt} \], where \( P(t) \) is the remaining substance at time \( t \), \( P_0 \) is the initial amount, and \( k \) is the decay constant.

The decay constant is found using \[ k = ext{ln}(1 - ext{decay rate}) \]. For strontium-90, with a decay rate of 2.5\%, you calculate \( k = ext{ln}(0.975) \).

This constant \( k \) shows how quickly or slowly the substance is decaying. A higher value of \( k \) would indicate a faster decay process. Understanding this constant is crucial for making accurate predictions about the material's future activity.

Graphing exponential functions provides a visual representation of how quantities change over time. In the case of radioactive decay, such as the strontium-90 example, graphing provides insights into how the remaining percentage of a substance decreases.The function \[ P(t) = 100(0.975)^t \] describes the decay over time, starting completely intact at 100\% (\( P_0 = 100 \)). When graphed, this function forms a downward-sloping curve which gets closer to zero but never actually reaches it as time progresses.

Plotting \( P \) against \( t \) from time \( t=0 \) to 100 years, you would observe:

• A rapid decline at the start of the curve
• A slowing rate of decline as \( t \) increases
• After about 100 years, as per calculations, a small portion (around 7.72\%) of the original substance remains

This graphing approach helps illustrate the concept of half-life and the nature of exponential decay, making it easier to comprehend how radioactive substances diminish over time.