Exponential Decay (Carbon-14 dating)

In 1949 Willard Frank Libby (1908-1980) invented the technique of carbon dating. This is based upon the radioactive decay of the isotope C-14: Two types of carbon occur naturally in our environment - carbon-12, which is not radio active, and carbon-14, which is radioactive with a halflife of 5730 years. All living plant and animal tissue contains both types of carbon, always in the same ratio. (The ratio is one part of carbon-14 to [Maple Math] parts of carbon-12.) The dating procedure is based on the assumption that this ratio has been constant in the atmosphere for a very long time. It is maintained by living plants and animals as long as they are alive. When the organism dies, however, no new carbon is absorbed, the amount of carbon-12 does not change, and the amount of carbon-14 decreases exponentially. Therefore the level of radioactivity (in disintegrations per hour per gram of carbon) decreases exponentially over time, i.e. if N(t) denotes the level of radioactivity t years after the organism dies, then [Maple Math] .

Exercise 4a

The half-life of C-14 is 5730 years. This means that a sample of any size of C-14 will decay to 1/2 of its size in 5730 years. Assuming exponential decay, the constant of proportionality k must satisfy the equation [Maple Math] , i.e. the factor [Maple Math] by which the initial amount is multiplied when [Maple Math] is [Maple Math] .

Exercise 4b

Exercise 5