Answer:
46.64 hours
Explanation:
The amount, A(t) of radioactive material with a half-life of t(0.5) remaining after time t is modeled using the formula below:
[tex]A(t)=A_0\mleft(\frac{1}{2}\mright)^{\frac{t}{t_{(0.5)}}}[/tex]• A(t)=quantity of the substance remaining
,• A(0)=initial quantity of the substance
,• t=time elapsed
,• t(0.5)=half life of the substance
In our case:
• A(t)=350 mg
,• A(0)=500 mg
,• t = 24 hours
Substitute into the formula above to find the half-life, t(0.5).
[tex]\begin{gathered} 350=500\mleft(\frac{1}{2}\mright)^{\frac{24}{t_{1/2}}} \\ \text{Divide both sides by 500} \\ \frac{350}{500}=\mleft(\frac{1}{2}\mright)^{\frac{24}{t_{1/2}}} \end{gathered}[/tex]Take the natural logarithm of both sides.
[tex]\begin{gathered} \ln (\frac{350}{500})=\ln (\frac{1}{2})^{\frac{24}{t_{1/2}}} \\ \frac{24}{t(0.5)}\ln (\frac{1}{2})=\ln (\frac{350}{500}) \end{gathered}[/tex]Divide both sides by ln(1/2).
[tex]\begin{gathered} \frac{24}{t(0.5)}=\frac{\ln(\frac{350}{500})}{\ln(\frac{1}{2})} \\ \implies\frac{t(0.5)}{24}=\frac{\ln(\frac{1}{2})}{\ln(\frac{350}{500})} \end{gathered}[/tex]Multiply both sides of the equation by 24.
[tex]\begin{gathered} \implies t(0.5)=24\times\frac{\ln(\frac{1}{2})}{\ln(\frac{350}{500})} \\ t(0.5)=46.64\text{ hours} \end{gathered}[/tex]The half-life of the radioactive element is approximately 46.64 hours.
