2.7. APPLICATIONS 75
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0
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When the rate of change is negative, the process is called exponential decay. This is theprocess which governs radioactive substances. It is the same formula which results, onlythis time it is of the form A(t) = Pexp(−rt) where r > 0. Consider the same example onlythis time consider y = 10e−.01t .
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Exercise 2.7.3 Carbon 14 has a half life of 5730 years. This means that if you start witha given amount of it and wait 5730 years, there will be half as much left. Carbon 14 is as-sumed to be constantly created by cosmic rays hitting the atmosphere so that the proportionof carbon in a living organism is the same now as it was a long time ago. This is of coursean assumption and there is evidence it is not true but this does not concern us here. Whenthe living thing dies, it quits replenishing the carbon 14 and so that which it has decaysaccording to the above half life. By measuring the amount in the remains of the dead thingand comparing with what it had when it was alive, one can determine an estimate for howlong it has been dead. Suppose then you measure the amount of carbon 14 in some deadwood and find there is 1/3 the amount there would have been when it was alive. How longago did the tree from which the wood came die?
Let A(t) be the amount of carbon 14 in the sample and let A0 be the amount whenit died. Then A(t) = A0 exp(−rt) . By assumption .33A0 = A0 exp(−rt) and cancellingthe A0 one can solve for t as follows. ln(.33) = −rt. If I knew what r was, I could thensolve for t. The half life is 5730 and so .5 = exp(−r5730) and so ln(.5) =−r (5730) fromproperties of ln described above, − ln(1/2) = ln
((1/2)−1
)= ln(2) . Therefore, r = ln2
5730 =
1.2097× 10−4. To get this number, I just used the computer. As mentioned above ln hasbeen tabulated. Therefore, in the problem of interest, ln(.33) = −
(1.2097×10−4
)t and
so t = ln(.33)−1.2097×10−4 = 9164.8 years.
So how did they find the half life of carbon 14? Did Noah have a sample of recentlydead wood in the ark and make some measurements which he recorded in the Book of theLaw of Noah which were then compared to measurements made in the twentieth centuryusing chronology determined by Bishop Ussher to determine that exactly 5730 years hadpassed? Actually, this is not the way it was done. The half life was also not established bythe decree of omniscient scientists. When I was young, I was constantly asked to believe