SOLUTION: A certain radioactive isotope has a half-life of approximately 1300 milliseconds. How much time would be required for a given amount of this isotope to decay to 35% of that amount

Question 541809: A certain radioactive isotope has a half-life of approximately 1300 milliseconds. How much time would be required for a given amount of this isotope to decay to 35% of that amount? Use the formula A=Ie^kt where A is the amount at the time, I is the initial amount and k is the decay constant unique to the substance.
Thank you
Answer by nerdybill(7384) (Show Source): You can put this solution on YOUR website!
A certain radioactive isotope has a half-life of approximately 1300 milliseconds. How much time would be required for a given amount of this isotope to decay to 35% of that amount? Use the formula A=Ie^kt where A is the amount at the time, I is the initial amount and k is the decay constant unique to the substance.
.
Given:
A=Ie^kt
Use:
"A certain radioactive isotope has a half-life of approximately 1300 milliseconds."
to find the constant 'k':
Let x = initial amount
x/2 = half the initial amount
x/2 = xe^(k*1300)
dividing both sides by x:
1/2 = e^(k*1300)
ln(.5) = k*1300
ln(.5)/1300 = k
.
Now we can answer:
How much time would be required for a given amount of this isotope to decay to 35% of that amount?
Let x = initial amount
then
.35x = 35% of initial amount
our formula:
A=Ie^kt
.35x = xe^(t*ln(.5)/1300)
dividing both sides by x:
.35 = e^(t*ln(.5)/1300)
ln(.35) = t*ln(.5)/1300
1300*ln(.35) = t*ln(.5)
1300*ln(.35)/ln(.5) = t
1968.95 milliseconds = t
OR,
1.96 seconds = t

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