The latest high-tech equipment permits reliable results to be obtained even with microscopic samples.
One way to help them, without going to the proportion indicated in the solution, would be to write the rate of decay over each period of 65$ years as $r$: that is, if there are $x$ micrograms of Carbon 14 present in the preserved plant at time $t$ then after 65$ years pass, there will be $rx$ micrograms of Carbon 14 remaining in the preserved plant.
Suppose we have a preserved plant and that the plant, at the time it died, contained 10 micrograms of Carbon 14 (one microgram is equal to one millionth of a gram).
An essential characteristic of exponential functions is that their values change by equal factors over equal intervals, that is, if $f(x)$ is an exponential function and $b$ a fixed real number, then the quotient $$ \frac $$ always takes the same value, that is, it does not depend on the real number $x_0$.
When it comes to dating archaeological samples, several timescale problems arise.
For example, Christian time counts the birth of Christ as the beginning, AD 1 (Anno Domini); everything that occurred before Christ is counted backwards from AD as BC (Before Christ).
In order to estimate when there is one microgram of Carbon 14 remaining in the preserved plant to the nearest $\frac$ years, the method of part (d) can be employed again, this time over the interval from 190$ years to 055$ years.
Each time this calculation is iterated, the estimated period of time for when one microgram of Carbon 14 remains is cut in half.
Here is one example of an isochron, based on measurements of basaltic meteorites (in this case the resulting date is 4.4 billion years) [Basaltic1981, pg. Skeptics of old-earth geology make great hay of these examples.
For example, creationist writer Henry Morris [Morris2000, pg.
The Greeks consider the first Olympic Games as the beginning or 776 BC.
The Muslims count the Prophet’s departure from Mecca, or the Hegira, as their beginning at AD 662.