Question 1

4. A radioactive substance has an initial mass of 223 kg. It decays at a continuous rate of 25.6% per year.A. Find an equation that models the mass of the substance after t years.B. How long does it take for the substance to reach a mass of 145.842 kg? Be sure to use your equation in part A. Then show all your work for your solution.

Question 2

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

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