Modeling an exponential decay

Question

I am given the following problem:

The burial cloth of an Egyptian mummy is estimated to contain 59% of the carbon-14 it contained originally. How long ago was the mummy buried?

Is it wrong to calculate it as it follows? $$0.59=1 \cdot (1-0.5)^x$$

From that, I got 4361 years, but a colleague has 4396 as an answer. Did I make a mistake somewhere?

Thank you.

Edit: I used 5730 as C-14's half life.

Answer 1

I get the same answer. I assume that after $5730$ years 50% of the original $14C$ content will be left.

To determine the decay (d) per year I used the equation $d^{5730}=0.5\Rightarrow d=0.5^{\frac1{5730}}$

Therefore the equation is

$$\left(0.5^{\frac1{5730}} \right)^x=0.59$$

$$x\cdot \log\left(0.5^{\frac1{5730}}\right)=\log(0.59)\Rightarrow x=\frac{\log(0.59)}{\log\left(0.5^{\frac1{5730}}\right)}\approx 4361.7$$

Remark

We can use the following equation to find out which half-life your colleague has used: $$\frac{\log(0.59)}{\log(0.5^{1/x})}=4396\Rightarrow x\approx 5775$$

Answer 2

The half life of Carbon 14 is $5730$ years so the correct calculation is to first solve

$$0.59=0.5^x$$

which gives $x\approx 0.7612$ and then calculate $0.7612$ half lives i.e. $4361$ years.