So, objects older than that do not contain enough of the isotope to be dated.

One specific example of exponential decay is purified kerosene, used for jet fuel.

The kerosene is purified by removing pollutants, using a clay filter.

When an organism dies, the amount of 12C present remains unchanged, but the 14C decays at a rate proportional to the amount present with a half-life of approximately 5700 years.

This change in the amount of 14C relative to the amount of 12C makes it possible to estimate the time at which the organism lived.

In other words, this function takes in a number of years, t, as its input value and gives back an output value of the percentage of carbon-14 remaining.

So, if you were asked to find out carbon's half-life value (the time it takes to decrease to half of its original size), you'd solve for t number of years when in any remains will have broken down.

A fossil found in an archaeological dig was found to contain 20% of the original amount of 14C. I do not get the $-0.693$ value, but perhaps my answer will help anyway.

If we assume Carbon-14 decays continuously, then $$C(t) = C_0e^,$$ where $C_0$ is the initial size of the sample. Since it takes 5,700 years for a sample to decay to half its size, we know $$\frac C_0 = C_0e^,$$ which means $$\frac = e^,$$ so the value of $C_0$ is irrelevant.

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.

Example: The population of a fictional country Tresperica grows at an exponential rate.

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