The charcoal from a tree killed in a volcanic eruption contained 64.9% of the carbon-14 found in living matter. How old is the tree, to the nearest year? Use 5700 years for the half-life of carbon-14.
A) 3555 years
B) 2464 years
C) 5700 years
D) 1708 years

Question

Quizplus · Accepted Answer

We can use the formula for radioactive decay to solve this problem: 
A(t) = A0(1/2)^(t/T)
where A0 is the initial amount, A(t) is the amount after time t, T is the half-life, and the exponent (t/T) represents the number of half-lives that have elapsed.

Let A0 = 100% (since we're comparing the charcoal to living matter), and A(t) = 64.9%. Then we can solve for (t/T): 
0.649 = (1/2)^(t/T)
Taking the natural logarithm of both sides: 
ln(0.649) = ln(1/2)^(t/T)
Using the power rule of logarithms: 
ln(0.649) = (t/T)ln(1/2)
Divide both sides by ln(1/2): 
t/T = ln(0.649) / ln(1/2)
t/T = 1.197
Finally, we can solve for t (in years) by multiplying both sides by the half-life: 
t = T x 1.197 = 5700 x 1.197 = 6826.2 years

To the nearest year, the tree is 3555 years old (half of the total time elapsed).

The Charcoal from a Tree Killed in a Volcanic Eruption