Question

A substance decays according to A=A0e−0.028t, where t is in hours and A0 is the initial amount. Determine the half-life of the substance. Round your answer to two decimal places, if necessary.

Answer 1

Solution

- The decay of the substance is governed by the following function:

$A=A_0e^{-0.0028t}$

- We are asked to find the half-life of the substance.

- The half-life of the substance is the time it takes for the substance to decay from its original mass to half its original mass.

- Based on this definition, we can say that:

$A=\frac{A_0}{2}$

- Thus, we can find the half-life as follows:

$\begin{gathered} A=A_0e^{-0.0028t} \\ \\ \text{ Put }A=\frac{A_0}{2} \\ \\ \frac{A_0}{2}=A_0e^{-0.0028t} \\ \\ \text{ Divide both sides by }A_0 \\ \\ \frac{1}{2}=e^{-0.0028t} \\ \\ \text{ Take the natural log of both sides} \\ \\ \ln\frac{1}{2}=\ln e^{-0.0028t} \\ \\ \ln\frac{1}{2}=-0.0028t \\ \\ \text{ Divide both sides by -0.0028} \\ \\ \therefore t=\frac{1}{-0.0028}\ln\frac{1}{2} \\ \\ t=247.55256...\approx247.55\text{ hours} \end{gathered}$

Final Answer

The half-life is 247.55 hours