Question

The number of grams A of a certain radioactive substance present at time, in years from the present, this given by the formula A = 45e^-0.0045tb. What is half-life of this substance?c. How much will be around in 2500 years?

Answer 1

b.

To determine the half life we plug half the original amount in the expression given and solve for t:

$\begin{gathered} \frac{45}{2}=45e^{-0.0045t} \\ e^{-0.0045t}=\frac{45}{2\cdot45} \\ e^{-0.0045t}=\frac{1}{2} \\ \ln e^{-0.0045t}=\ln (\frac{1}{2}) \\ -0.0045t=\ln (\frac{1}{2}) \\ t=\frac{1}{-0.0045}\ln (\frac{1}{2}) \\ t=154.03 \end{gathered}$

Therefore the half-life of the substance is approximately 154 years.

c.

To determine how much of the substance will be after 2500 years we just plug this value in the expression:

$\begin{gathered} A=45e^{-0.0045\cdot2500} \\ A=5.85\times10^{-4} \end{gathered}$

Therefore after 2500 years there will be:

$5.85\times10^{-4}\text{ gr}$

of the substance.