SOLUTION: please help: Suppose you have 500 grams of a radioactive element whose half life is 72 years. a) Find the function for the quantity, Q, left in t years. b) Find to the neare

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: please help: Suppose you have 500 grams of a radioactive element whose half life is 72 years. a) Find the function for the quantity, Q, left in t years. b) Find to the neare      Log On


   

Question 331902: please help:
Suppose you have 500 grams of a radioactive element whose half life is 72 years.
a) Find the function for the quantity, Q, left in t years.
b) Find to the nearest tenth of a year when there will be 100 grams left.
Answer by nerdybill(7384) About Me  (Show Source):
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Suppose you have 500 grams of a radioactive element whose half life is 72 years.
a) Find the function for the quantity, Q, left in t years.
b) Find to the nearest tenth of a year when there will be 100 grams left.
.
Exponential decay/growth formula:
N = Noe^(rt)
where
N is amount after t time
No is the initial amount
r is rate of decay
t is years
.
Find r from "500 grams of a radioactive element whose half life is 72 years":
N = Noe^(rt)
.5(500) = 500e^(r*72)
.5 = e^(r*72)
ln(.5) = r*72
ln(.5)/72 = r
.
Replacing r with the above in:
N = Noe^(rt)
To get:
N = 500e^(tln(.5)/72)
which ANSWERS part a.
.
Part b:
b) Find to the nearest tenth of a year when there will be 100 grams left.
Starting with:
N = 500e^(tln(.5)/72)
Replace N with 100 and solve for t
100 = 500e^(tln(.5)/72)
.2 = e^(tln(.5)/72)
ln(.2) = tln(.5)/72
72ln(.2) = tln(.5)
72ln(.2)/ln(.5) = t
167.2 years = t