Exponential growth problem

Question

Assume you find a mammoth tooth and want to know how old it is; you use the carbon 14 method. Carbon 14 has a half life of 5730 years. Youth tooth has only 4% of the count of a fresh tooth. How old is the tooth you found?

Answer 1

You use A = Pe^(rt). This is normally the formula for money  and compound interest but it is used for all growth and decay. The trick with half life is that no matter what you started with, P, A is always half as much let's say you started with 70. In 5730 years you will have 35. So

35 = 70*e^(5730r)    divide by 70

.5 = e^(5730r) Notice if you used 250 and 125, you would get.5. if you sed 6000 and 3000, you'll get .5. Take Ln of both sides

Ln .5 = 5730 *r

(Ln .5)/5730 = r = -.0001209680943

 

Plug in for r

A = Pe^( -.0001209680943 t) Now instead of being left with 50%, you're left with 4% of the original. So

.04 = e^( -.0001209680943 t)   take Ln of both sides 

Ln .04 = -.0001209680943 t divide and get 

26,609.25958..........so it happened during the 26,610th year.