 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that A paleontologist examining the bones of a prehistoric animal estimates that they contain 2% as much carbon-14 as they would have contained when the animal was alive. How long ago the animal lived given that the amount of carbon-14 present in the animal after T years is modelled by the function y is equal to A into e raised to power minus of 0.00012 into T. Now let us start with the solution of the given question. In this question we are given the exponential decay model of carbon-14 which is given by y is equal to A into e raised to power minus of 0.00012 into T. It is given that the bones of prehistoric animal contain 2% as much carbon-14 as they would have contained when the animal was alive and we have to find how long ago did the animal live in the given function that is y is equal to A into e raised to power minus of 0.00012 into T. A is the initial amount of carbon-14 in the animal's body and the amount y that remains after T years is given as 2% of A that is y is equal to 2% of initial amount A that is 2 by 100 into A which is equal to 0.02 into A. So we put y is equal to 0.02A in the given function we get 0.02A is equal to A into e raised to power minus of 0.00012 into T. Now we divide by A on both sides of the equation and we get 0.02 is equal to e raised to power minus of 0.00012 into T. Since the base is e so we take natural logarithm on both sides of the equation and we get natural log of 0.02 is equal to natural log of e raised to power minus of 0.00012 into T and we know that natural log of e raised to power A is equal to A. Now using this result this implies that natural log of 0.02 is equal to minus of 0.00012 into T. Now we divide both sides by minus of 0.00012 into T. and we get natural log of 0.02 upon minus of 0.00012 is equal to minus of 0.00012 into T upon minus of 0.00012 which implies that natural log of 0.02 upon minus of 0.00012 is equal to T. Using scientific calculator we get the value of natural log of 0.02 is equal to minus of 3.91202 which implies that natural log of 0.02 can be written as minus of 3.91202 upon minus of 0.00012 is equal to T which further implies that T is approximately equal to 32,600 thus the animal lived about 32,600 years ago which is the required answer. This completes our session. Hope you enjoyed this session.