Given a radioactive substance, experiments have found that its decay rate is a c

Question

Given a radioactive substance, experiments have found that its decay rate is a constant proportion of the amount of the substance. Suppose we have 100 grains of radioactive Carbon-14, ^14C. Write down a differential equation initial value problem that models the mass of^14C after t years. The half-life of^14C is about 5730 years. Find the mass that remains after t years by solving the initial value problem in part (a). When a plant or animal dies, it stops replacing its carbon and the amount of^14C begins to decrease through radioactive decay. Suppose an archaeological discovery revealed a papyrus fragment that had about 70% as much^14C as does the plant material on the earth today. Estimate the age of the papyrus.

Explanation / Answer

Solution : decay rate is proportional to the substance

Let X (T) be the amount of the substance .hence d [ X (T)] / dT= - k X [ - ve sign for decaying

solving the DE using variable seperable method d X / X = - kdT . integrating on either side

log X = - k T + log c where c is a constant

   X = e - kT  c ------(1)

   to evaluate c let us use the half time period x =50 and t= 5730 and when t =0 X =100

   substituting in (1) 100 = 1 c and 50 = e - k 5730 (100) => log 1 /2 = - k (5730)

. 3010 / 5730 =k

     

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