A Sample of Tritium-3 Decayed to 94.5% of its Original Amount After a Year. (a) What is the half-life of tritium-3? Please use the following notation when explaining the solution. The formula I am aware of is $P(t)=P_oe^{kt}$ $P_o=\ initial\ amount$ t = time k = growth rate (a) What is the half-life of tritium-3? How would I calculate? (b) How long would it take the sample to decay to 20% of its original amount? > Please keep in mind I am looking for clear and suitable methods a 2nd semester Calculus student can use to solve problems such as this.

Notice, $$P(t)=P_{0}e^{kt}$$ Where, growth rate $$k=-\lambda=-\frac{\ln 2}{t_{1/2}}$$ Since, the amount after time $t=1\ year$ decays to $P(t)=94.5$ % of initial value $P_0$, hence substituting the corresponding values in the formula as follows $$\frac{94.5}{100}P_0=P_0e^{-\lambda \times 1}$$ $$e^{-\lambda}=0.945$$ $$\lambda=-\ln(0.945)$$ Hence the half life of tritium-3 is given $$t_{1/2}=\frac{\ln 2}{-\ln(0.945)}=12.25\ \text{years}$$

Now, the time $t$ taken by the sample to decay to $20$ % of original amount hence, $$\frac{20}{100}P_0=P_0e^{-\lambda t}$$ $$e^{-\lambda t}=0.2$$ $$-\lambda t=\ln(0.2)$$ $$t=\frac{\ln(0.2)}{\ln(0.945)}=28.45 \ \text{years}$$