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An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. This type of function is used to model scenarios where quantities grow or decay at a rate proportional to their current value. Radioactive decay is a classic example of an exponential decay process.In our problem with cesium-137, the mass of the substance decreases over time, following the model:

• \( A(t) = A_0 \, e^{kt} \)

The term \( A(t) \) represents the mass of the radioactive substance at time \( t \), while \( A_0 \) is the initial mass, which is 10 grams in our case. The variable \( k \) is the decay constant, which we must determine using the known half-life of the substance. By solving this exponential equation, we can predict how much mass remains at any given time.

The concept of half-life is fundamental to understanding radioactive decay. It is defined as the time required for a quantity to reduce to half of its initial value. For cesium-137, the half-life is 30 years, meaning every 30 years, the mass of cesium-137 decreases to half its size.Half-life is crucial because it allows us to calculate the decay constant \( k \). The relationship between half-life and the decay constant is given by the formula:

• \( k = \frac{\ln(0.5)}{t_{1/2}} \)

This logarithmic expression ensures that as time progresses, the amount of substance depletes at a consistent rate. Knowing the half-life enables us to determine the rate at which the material decays, making it easier to model future scenarios.

The decay constant \( k \) is a crucial part of the exponential decay formula and is specific to each radioactive material. It defines the rate at which the material decays and is directly linked to the half-life.To find \( k \) for cesium-137, we use the equation:

• \( k = \frac{\ln(0.5)}{30} \)

Here, the natural logarithm of 0.5 (representing a reduction to half the initial amount) is divided by the half-life, which is 30 years for cesium-137. This calculation provides us with \( k \), allowing us to fully utilize the exponential function to determine how the mass changes over any given period.Understanding \( k \) equips us to accurately model the decay process and solve for scenarios where we need to find the mass remaining after certain periods or discern time periods for a set mass. This concept is essential for predicting decay over both short and long spans effectively.