Answer:
See explanation below.
Explanation:
For this case we atart fom the proportional model given by the following differential equation:
And if we rewrite this expression we got:
If we integrate both sides we got:
And using exponential on both sides we got:
Where represent the initial amount for the isotope and t the time in years and A the amount remaining.
If we want to apply a model for the half life we know that after some time definfd the amount remaining is the hal, so if we apply this we got:
We can cancel and we got:
If we solve for k we can apply natural log on both sides and we got:
And that would be our proportional constant on this case.
If we replace this value for k int our model we will see that:
And using properties of logs we can rewrite this like that:
And thats the common formula used for the helf life time.