Question

A radioactive substance decays according to A=A0e−0.0028t, where A0 is the initial amount and t is the time in years. If A0=710 grams, find the time for the radioactive substance to decay to 366 grams. Round your answer to two decimal places, if necessary.

Answer 1

Solution

- The function given is:

$A=A_0e^{-0.0028t}$

- We have been given:

$\begin{gathered} A_0=710g \\ A=366g \end{gathered}$

- We are required to find the time for the radioactive substance to decay. That means we need to find the value of t in the equation.

- Thus, we have that:

$\begin{gathered} 366=710e^{-0.0028t} \\ \text{ Divide both sides by 710} \\ \frac{366}{710}=e^{-0.0028t} \\ \\ \text{ Take the natural log of both sides} \\ \ln(\frac{366}{710})=\ln(e^{-0.0028t}) \\ \\ \ln(\frac{366}{710})=-0.0028t \\ \\ \text{ Divide both sides by -0.0028} \\ \\ \therefore t=\frac{1}{-0.0028}\ln(\frac{366}{710}) \\ \\ t=236.6541...\approx236.65\text{ years} \end{gathered}$

Final Answer

The answer is 236.65 years