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# How many years old is the artifact?

Question. A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of 1.15% per day.

A wooden artifact from an archeological dig contains 60 percent of the carbon-14 that is present in living trees. To the nearest year, about how many years old is the artifact? (The half-life of carbon-14 is 5730 years.)

How to find and solve this question?

I don't know how to calculate and to find this correct formula?

### 2 Answers

- PuzzlingLv 74 weeks ago
I'm assuming you are asking about the half-life problem and we should ignore the iodine-125 part?

First, let's think a little about what we expect in terms of a solution.

The half life is 5730 years. So after 5730 years, we would expect to have 50 percent of the carbon-14. We instead have 60 percent, so it hasn't gone through a full half life period. The answer will be less than 5730 years, right?

The formula for the amount of material after t years is:

A(t) = Ao * (1/2)^(t/λ)

Ao : Initial amount

A(t) : Amount at time t (in years)

t : time (in years)

λ : half-life (in years)

So for your case, you have:

0.6 * Ao = Ao * (1/2)^(t/5730)

You can cancel the original amount from both sides:

0.6 = (1/2)^(t/5730)

Now take the log of both sides:

log(0.6) = log((0.5)^(t/5730))

Use this rule of logs --> log(a^b) = b log(a)

log(0.6) = t/5730 * log(0.5)

Solve for t:

t = 5730 log(0.6) / log(0.5)

I'll leave the final calculation to you. Round accordingly.

- BryceLv 74 weeks ago
0.50= e^(5730k)

k=ln0.50 /5730≈ -0.000121

0.60= e^(-0.000121t)

t= ln0.60 /-0.000121≈ 4223 years