To model the decay of Iodine-125 in the tumor, we start by using the exponential decay formula. The formula used to represent exponential decay is:
[tex]\[ A(t) = A_0 \cdot \exp(-k \cdot t) \][/tex]
Where:
- [tex]\( A(t) \)[/tex] is the amount of Iodine-125 remaining after [tex]\( t \)[/tex] days.
- [tex]\( A_0 \)[/tex] is the initial amount of Iodine-125, which is [tex]\( 0.5 \)[/tex] grams.
- [tex]\( k \)[/tex] is the decay rate per day.
- [tex]\( t \)[/tex] is the time in days.
- [tex]\( \exp \)[/tex] represents the exponential function, commonly written [tex]\( e^{(x)} \)[/tex].
Given the decay rate [tex]\( k = 1.15\% \)[/tex] per day, we convert this percentage to a decimal for use in our equations:
[tex]\[ k = 1.15\% = 0.0115 \][/tex]
Thus, the exponential model representing the amount of Iodine-125 remaining in the tumor after [tex]\( t \)[/tex] days is:
[tex]\[ A(t) = 0.5 \cdot \exp(-0.0115 \cdot t) \][/tex]
To find the amount of Iodine-125 remaining after 60 days, we substitute [tex]\( t = 60 \)[/tex] into the model:
[tex]\[ A(60) = 0.5 \cdot \exp(-0.0115 \cdot 60) \][/tex]
Therefore, the exact amount of Iodine-125 remaining in the tumor after 60 days is:
[tex]\[ A(60) = 0.2507880345330278 \text{ grams} \][/tex]
This concludes our step-by-step solution showing how to model the decay and determine the remaining amount after a specified time period.
