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Sagot :
We are asked to determine the decay constant of a radioactive element. To do that we will use the following formula:
[tex]A=A_0e^{-kt}[/tex]Where:
[tex]\begin{gathered} A=\text{ quantity of the element} \\ A_0=\text{ initial quantity} \\ k=\text{ decay constant} \\ t=\text{ time} \end{gathered}[/tex]The half time is the time when the quantity of the element is half the initial quantity. Therefore, we have:
[tex]\frac{A_0}{2}=A_0e^{-kt}[/tex]Now, we cancel out the initial quantitu:
[tex]\frac{1}{2}=e^{-kt}[/tex]Now, we solve for "t". First, we take the natural logarithm to both sides:
[tex]\ln(\frac{1}{2})=-kt[/tex]Now, we divide both sides by -t:
[tex]-\frac{1}{t}\ln(\frac{1}{2})=k[/tex]Now, we plug in the value of the time:
[tex]-\frac{1}{0.44day}\ln(\frac{1}{2})=k[/tex]Solving the operations:
[tex]1.575\frac{1}{day}=k[/tex]Therefore, the decay constant is 1.575 1/day.
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