Chapter 9.1, Problem 14E

The carbon in living matter contains a minute proportion of the radioactive isotope C−14. This radiocarbon arises from cosmic−ray bombardment in the upper atmosphere and enters living systems by exchange processes. After the death of an organism, exchange stops, and the carbon decays. Therefore, carbon dating enables us to calculate the time at which an organism died. Let $x(t)$ be the proportion of the original C−14 still present t years after death. By definition, $x(0)=1=100\%$ . We are told that $x(t)$ satisfies the differential equation
   $\frac{dx}{dt}=-\frac{1}{8270}x$
a. Find a formula for $x(t)$. Determine the half−life of C−14 (that is, the time it takes for half of the C−14 to decay).
b. The Iceman. In 1991, the body of a man was found in melting snow in the Alps of Northern Italy. A well−known historian in Innsbruck, Austria, determined that the man had lived in the Bronze Age, which started about 2000 B.C. in that region.

Examination of tissue samples performed independently at Zürich and Oxford revealed that 47% of the C−14 present in the body at the time of his death had decayed. When did this man die? Is the result of the carbon dating compatible with the estimate of the Austrian historian?