Radiocarbon Dating - Calculus, Growth Functions

Question

(11) Scientists can determine the age of ancient objects by the method of radiocarbon dating. The bombardment of the
upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, C, with a half-life of about
5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates C through food
chains. When a plant or animal dies, it stops replacing its  carbon and the amount of C begins to decrease through
radioactive decay. Therefore the level of radioactivity must also decay exponentially.

 

A parchment fragment was discovered that had about 74% as much C radioactivity as does plant material on the earth today. Estimate the age of the parchment.

Answer 1

Let A(t) = amount remaining in t years

 

     A(t) = A₀e^kt, where A₀ is the initial amount and k is a constant to                be determined

 

We know that A(5730) = 0.5A₀, so A₀e^5730k = 0.5A₀

 

                                                    e^5730k = 0.5

 

                                                      5730k = ln(0.5)

 

                                                            k = -0.000121

 

A(t) = A₀e^-0.000121t

 

So, if A(t) = 0.74A₀, then 0.74A₀ = A₀e^-0.000121t

 

                                      0.74 = e^-0.000121t

 

                                    ln(0.74) = -0.000121t

 

                                         t = 2488.4718

 

The parchment is about 2488 years old.