Scientist can determine the age of ancient objects by a method called radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, 14C, with a half-life of about 5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates 14C through food chains. When a plant or animal dies, it stops replacing its carbon and the amount of 14C begins to decrease through radioactive decay. Therefore, the level of radioactivity must also decay exponentially. A parchment fragment was discovered that had about 74% as much 14C radioactivity as does plant material on Earth today. Estimate the age of the parchment. (Round your answer to the nearest hundred years.) yr

We know half-life is 5730 years and that the parchment has retained 74% of its Carbon-14. For [tex]A_{0[/tex] let's just assume that there are 100 original atoms of Carbon-14 and for A(t) let's assume there are 74 Carbon-14 atoms AFTER the amount of time has passed. That way, 74% of the C-14 still remains as 74/100 is 74%. Not quite sure how to explain it but I hope you get it. h is the last variable we need to know and it's just the half-life, which has been given to us already, 5730 years, so now we have this.

[tex]74=100(1/2)^t^/^5^7^3^0[/tex]

Now, solve. First, divide by 100.

[tex]0.74=(0.5)^t^/^5^7^3^0[/tex]

Take the log of everything

[tex]log(0.74)=\frac{t}{5730} log(0.5)[/tex]

Divide the entire equation by log (0.5) and multiply the entire equation by 5730 to isolate the t and get

[tex]5730\frac{log(0.74)}{log(0.5)} =t[/tex]

Use your calculator to solve that giant mess for t and you'll get that t is roughly 2489.128182 years. Round that to the nearest hundred years, and you'll find the hopefully correct answer is 2500 years.

Really hope that all the equations that I wrote came out good and that that's right, this is definitely the longest answer I've ever written.