To determine the amount of a radioactive substance remaining after a certain period of time, we use the concept of half-life. The half-life of a substance is the time required for half of the substance to decay.
Given:
- The half-life of Americium: 25 minutes
- Initial amount of Americium: [tex]\( A_0 \)[/tex]
The formula for the remaining amount of a substance after time [tex]\( t \)[/tex] based on its half-life is:
[tex]\[ A(t) = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T}} \][/tex]
where:
- [tex]\( A(t) \)[/tex] is the amount remaining after time [tex]\( t \)[/tex]
- [tex]\( A_0 \)[/tex] is the initial amount
- [tex]\( t \)[/tex] is the elapsed time
- [tex]\( T \)[/tex] is the half-life of the substance
Using the given half-life [tex]\( T = 25 \)[/tex] minutes and the elapsed time [tex]\( t \)[/tex]:
[tex]\[ A(t) = A_0 \left( \frac{1}{2} \right)^{\frac{t}{25}} \][/tex]
Thus, the correct expression for the amount of Americium remaining after [tex]\( t \)[/tex] minutes is:
[tex]\[ A(t) = A_0 \left( \frac{1}{2} \right)^{\frac{t}{25}} \][/tex]
Therefore, the correct choice among the options provided is:
(1) [tex]\( A_0\left(\frac{1}{2}\right)^{\frac{t}{25}} \)[/tex]
