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Understanding **half-life** is crucial when studying radioactive decay. Half-life is the amount of time required for half of a radioactive substance to decay. By knowing the half-life, you can predict how quickly a material will lose its radioactivity.

For example, in the problem, cobalt-60 has a half-life of 5.3 years. This means that every 5.3 years, only half of the original amount of cobalt-60 remains radioactive.

The concept of half-life helps us determine how long a radioactive element will stay active, which is vital in fields like medicine, archaeology, and nuclear power.

To remember, shorter half-lives mean a substance decays faster, while longer half-lives mean it decays more slowly.

The **decay constant**, denoted by the Greek letter λ, is a number that gives the rate of decay for a radioactive substance. It is closely related to the half-life.

The decay constant can be calculated using the half-life with the formula:
\(\lambda = \frac{\ln(2)}{t_{1/2}}\)

In this formula, \(\ln(2)\) is the natural logarithm of 2 (about 0.693), and \(t_{1/2}\) is the half-life of the substance.

For cobalt-60 with a half-life of 5.3 years, substituting into the formula we get:
\[ \lambda = \frac{\ln(2)}{5.3} \approx 0.1307 \text{ per year} \]

The decay constant provides a straightforward way to quantify how quickly a radioactive material decays over time, making it indispensable in calculations involving exponential decay.

When a radioactive substance diminishes over time, it follows a pattern known as **exponential decay**. This pattern means that the quantity decreases at a rate proportional to its current value.

In mathematical terms, exponential decay can be represented by the equation:
\[ N(t) = N_0 e^{-\lambda t} \]

Here, \(N(t)\) is the quantity of the substance at time \(t\), \(N_0\) is the initial quantity, \(e\) is the base of the natural logarithm, and \(\lambda\) is the decay constant.

This equation shows how the quantity of a substance decreases over time. As \(t\) increases, \(e^{-\lambda t}\) gets smaller, leading to a reduction in \(N(t)\).

Exponential decay is a key concept not just in physics, but also in fields like pharmacology and population biology, where similar decay processes are observed.