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It might take a millisecond, or it might take a century. But if you have a large enough sample, a pattern begins to emerge.

It takes a certain amount of time for half the atoms in a sample to decay.

Now, take the logarithm of both sides to get $$-0.693 = -5700k,$$ from which we can derive $$k \approx 1.22 \cdot 10^.$$ So either the answer is that ridiculously big number (9.17e7) or 30,476 years, being calculated with the equation I provided and the first equation in your answer, respectively.

A useful application of half-lives is radioactive dating.It has been determined that the rate of radioactive decay is first order.We can apply our knowledge of first order kinetics to radioactive decay to determine rate constants, original and remaining amounts of radioisotopes, half-lives of the radioisotopes, and apply this knowledge to the dating of archeological artifacts through a process known as carbon-14 dating., where r is a measurement of the rate of decay, k is the first order rate constant for the isotope, and N is the amount of radioisotope at the moment when the rate is measured.The rate of decay is often referred to as the activity of the isotope and is often measured in Curies (Ci), one curie = 3.700 x 10" is the initial amount of radioisotope at the beginning of the period, and "k" is the rate constant for the radioisotope being studied.

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