[tex]\begin{gathered} \text{ Given} \\ A=223\operatorname{kg}\text{ as the initial mass} \\ r=-0.0256\text{ as the rate, converted from percentage, we use negative since it decays} \\ t=\text{time in years} \\ \text{ So we use the exponential form } \\ M(t)=Ae^{rt}\text{ (I used M for mass)} \\ \text{ A. Equation is }M(t)=Ae^{rt},\text{ since r=-0.0256}\rightarrow M(t)=Ae^{-0.0256t} \\ \text{ Using our equation in letter A, subsitute with }M(t)=145.842\operatorname{kg}\text{ after} \\ \text{some t years} \\ M(t)=Ae^{rt} \\ 145.842=(223)e^{(-0.0256)t} \\ \frac{145.842}{223}=\frac{(\cancel{223})e^{(-0.0256)t}}{\cancel{223}} \\ \frac{145.842}{223}=e^{-0.0256t}\text{ get the natural log of both sides} \\ \ln (\frac{145.842}{223})=-0.0256t \\ \frac{\ln(\frac{145.842}{223})}{-0.0256}=\frac{\cancel{-0.0256}t}{\cancel{-0.0256}} \\ t=\frac{\ln(\frac{145.842}{223})}{-0.0256}\text{ input this in a calculator and we get} \\ t\approx16.58780 \\ \text{ Therefore it would take around 16.587 years to get to 145.842kg} \end{gathered}[/tex]
