Archaeologists can determine the age of artifacts made of wood or bone by measuring the amount of the radioactive isotope 14C present in the object. The amount of isotope decreases in a first-order process. If 14.5% of the original amount of 14C is present in a wooden tool at the time of analysis, what is the age of the tool? The half-life of 14C is 5,730 yr. Give your answer in scientific notation.
.5^n = .145
n = log(.145) / log(.5) = 2.786
2.786 * 5730 = 1.60 x 10^4
.145 = (1/2)^n
log .145 = n log .5
n = 2.786 half lives
2.786 * 5730 = 15963
= 1.5963*10^4 years
To determine the age of the wooden tool, we can use the concept of half-life. The half-life of a radioactive isotope is the time it takes for half of the original amount of the isotope to decay.
In this case, the half-life of 14C is given as 5,730 years. This means that after 5,730 years, half of the original 14C would have decayed.
Since the amount of 14C in the wooden tool is currently 14.5% of the original amount, it means that 85.5% of the isotope has decayed. This corresponds to one half-life.
To find the age of the wooden tool, we need to determine how many half-lives have passed. We can do this by dividing the percentage of remaining isotope (14.5%) by the percentage of isotope decayed (85.5%):
Number of half-lives = log(14.5% / 85.5%) / log(0.5)
Using the natural logarithm (ln) for the log function, we get:
Number of half-lives = ln(0.145 / 0.855) / ln(0.5)
Calculating this gives us approximately 0.574.
Now, we can find the age of the tool by multiplying the number of half-lives by the half-life of 14C:
Age of the tool = Number of half-lives * half-life of 14C
= 0.574 * 5730 years
≈ 3292 years
Therefore, the age of the wooden tool is approximately 3292 years.
Expressed in scientific notation, the age of the tool would be 3.292 × 10^3 years.