Question 226392: You have just isolated a new radioactive element. If you can determine its
half-life, you will win the Nobel Prize in physics. You purify a sample of 2 grams. One of
your colleagues steals half of it, and three days later you find that 0.1 gram of the
radioactive material is still left. Find an exponential model for this problem in the form
!
A(t) = P*e^k-t and find the half-life for this element?
Answer by nerdybill(7384)   (Show Source):
You can put this solution on YOUR website! You purify a sample of 2 grams. One of your colleagues steals half of it, and three days later you find that 0.1 gram of the radioactive material is still left. Find an exponential model for this problem in the form
A(t) = P*e^kt and find the half-life for this element?
.
Since your colleague stole half (of the 2 grams), you started with1 gram so
P = 1 gram
.
Three days later you have 0.1 gram:
t = 3 days
A(t) = 0.1 grams
.
Solve for "rate of growth" (k):
A(t) = P*e^kt
0.1 = 1*e^(3k)
0.1 = e^(3k)
ln(0.1) = 3k
ln(0.1)/3 = k
-0.76752836433134856133933048489479 = k (keeping all your significant figures help in finding a more accurate result later)
.
Our "exponential model" is:
A(t) = 1*e^(tln(0.1)/3)
A(t) = e^(tln(0.1)/3)
A(t) = e^(-0.767528t)
.
Now to find "half-life", set A(t) to the "half-life" amount .5 (since we started with 1 gram) and solve for t:
A(t) = e^(-0.767528t)
.5 = e^(-0.767528t)
ln(.5) = -0.767528t
ln(.5)/-0.767528 = t
0.9 days = t
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