SOLUTION: The half-life of carbon 14 is 5,730 years. Approximately, how old is a bone that has 70% of its original carbon 14?

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Question 1142059: The half-life of carbon 14 is 5,730 years. Approximately, how old is a bone that has 70% of its original carbon 14?
Found 2 solutions by Theo, greenestamps:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the formula to use is f = p * e ^ (r * t)

f is the future value
p is the present value
r is the interest rate per time period
t is the number of time periods

if you are looking for the half life, then f = 1/3 and p = 1.

the p becomes silent and the formula becomes 1/2 = e ^ (r * t)

the time is 5730 years.

the formula becomes 1/2 = e ^ (r * 5730)

take the natural log of both sides of the equation to get:

ln(1/2) = ln(e ^ (r * 5730))

since ln(e ^ (r * 5730)) = r * 5730 * ln(e) and, since ln(e) = 1, the formula becomes:

ln(1/2) = r * 5730

divide both sides of this equation by 5730 to get:

ln(1/2) / 5730 = r

solve for r to get:

r = -.000120968094

to confirm this is correct, replace r with that in the original equation to get:

f = e ^ (-.000120968094 * 5730)

solve for f to get:

f = 1/2

that confirms the formula is correct.

the question is...

how old is a bone that has 70% of its original carbon 14?

the formula which was .5 = e^ ( r * t) becomes:

.70 = e ^ (r * t)

when r = -.000120968094, the formula becomes:

.70 = e ^ (-.000120968094 * t)

you take natural log of both sides of this equation and simplify it to get:

ln(.70) = -.000120968094 * t

divide both sides of the equation by -.000120968094 and solve for t to get:

t = ln(.70) /-.000120968094 = 2948.50428 years.

once you find r, you can graph the equation.

it looks like this.

$$$










Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The fraction remaining after n half-lives is %281%2F2%29%5En.

You want to know when the fraction remaining is 70%, or .7.

%281%2F2%29%5En+=+%28.5%29%5En+=+.7

The variable is an exponent, so use logarithms:

n%2Alog%28.5%29+=+log%28.7%29
n+=+log%28.7%29%2Flog%28.5%29 = 0.514573 to 6 decimal places.

Then the age is the number of half-lives, multiplied by the number of half-lives.

5730%2A0.514573+=+2948.5

Since the problem says approximately, a good answer is 2950 years.