Half life means the amount of carbon-14 would 1/2 of what the original amount was.
Since the half life takes 5715 years, part a is asking how much there would be after 5100, so not a full half life.
First find the percentage of the number of years compared to the half life:
5100 / 5715 = 0.89 = 89%
The original amount was 50 grams, so the half life would be 25 grams.
So you need to calculate 89% of the half life: 25 * 0.89 = 22.3 grams
Now subtract that from the starting amount: 50 - 22.3 = 27.7 grams are left after 5100 years.
( A little more than half, because 5100 years is less than the 5715 years)
B. every 5715 years the amount is cut in half, so from 50 to 25, then another 5715 years to cut in half from 25 to 12.5. etc.
To find the amount of time we would set up an exponential decay formula:
remaining quantity = original quantity x 1/2 ^ time / half life
10 = 50*1/2 ^ X/5715, when solving for x because it is part of the exponent it becomes a logarithm problem.
10 = 50*2^(x/5715)
10 * 2^(x/5715) = 50
2^(x/5715) = 5
LN(2^(x/5715)) = ln5
x = 5715ln(5) / ln2
x = 13,269.82
Rounded to nearest whole year = 13,270 years.
