Answer:
[tex]\displaystyle k = \frac{1}{32}\ln\frac{1}{2} \approx -0.02166[/tex]
Step-by-step explanation:
The decay formula is given by:
[tex]\displaystyle P(t) = P_0e^{kt}[/tex]
Where k is some constant, P₀ is the initial population, and t is the number of years.
Because the substance has a half-life of 32 years, P(t) = 1/2P₀ when t = 32. Substitute:
[tex]\displaystyle \frac{1}{2}P_0 = P_0 e^{k(32)}[/tex]
Solve for k:
[tex]\displaystyle \begin{aligned} \frac{1}{2} & = e^{32k} \\ \\ \ln\left(e^{32k}\right) & = \ln\left(\frac{1}{2}\right) \\ \\ 32k & = \ln\frac{1}{2} \\ \\ k & = \frac{1}{32}\ln\frac{1}{2} \\ \\ &\approx -0.02166\end{aligned}[/tex]
In conclusion, the value of k is about -0.02166.
