Question

complete the table shown to the right for the half - life of a certain radioactive substance.
half - life
decay rate, k
4.8% per year = - 0.048
the half - life is (square) years.
(round to one decimal place as needed.)

Explanation:

Step1: Recall half - life formula

For exponential decay, the formula relating the decay rate \(k\) and the half - life \(t_{1/2}\) is \(t_{1/2}=\frac{\ln(2)}{|k|}\). Here, the decay rate \(k=- 0.048\) (since it's a decay rate of \(4.8\%=0.048\) per year), so \(|k| = 0.048\).

Step2: Calculate the half - life

Substitute \(|k| = 0.048\) into the formula \(t_{1/2}=\frac{\ln(2)}{0.048}\). We know that \(\ln(2)\approx0.6931\). Then \(t_{1/2}=\frac{0.6931}{0.048}\approx14.4\) (rounded to one decimal place).

Answer:

\(14.4\)