Exercise 3 Solution

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The exponential model for A(t), the amount of material at time t, gives the decay behavior,

where A₀ is the initial amount of material (that is, the amount at t = 0), and r is the decay constant.

We are given that

Thus,

or,

Solving for r by taking the logs (natural log, ln) of both sides gives,

Solving for r gives,

So this radioactive material obeys the exponential model,

To find the half-life, we need to find the time t when half the material is gone—that is, when

Substituting and solving for t gives,

Thus, the half-life is 78.9 years. Note that we never needed to know the initial amount of material (A₀) to solve this problem.