To determine how much carbon-14 remains in a fossilized tree branch after 50,000 years, we will use the decay formula for radioactive substances:
[tex]\[ N = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T}} \][/tex]
where:
- [tex]\( N \)[/tex] is the remaining amount of the substance after time [tex]\( t \)[/tex].
- [tex]\( N_0 \)[/tex] is the initial amount of the substance.
- [tex]\( t \)[/tex] is the time that has passed.
- [tex]\( T \)[/tex] is the half-life of the substance.
Given:
- [tex]\( N_0 = 430 \)[/tex] grams (initial amount of carbon-14)
- [tex]\( T = 5730 \)[/tex] years (half-life of carbon-14)
- [tex]\( t = 50,000 \)[/tex] years (time period)
First, we will calculate the number of half-lives that have passed during the time period:
[tex]\[
\frac{t}{T} = \frac{50,000}{5730} \approx 8.726
\][/tex]
Next, we apply this value to the decay formula:
[tex]\[
N = 430 \left(\frac{1}{2}\right)^{8.726}
\][/tex]
Evaluating the exponent:
[tex]\[
\left(\frac{1}{2}\right)^{8.726} \approx 0.002361627
\][/tex]
Therefore,
[tex]\[
N = 430 \times 0.002361627 \approx 1.015
\][/tex]
After 50,000 years, the remaining amount of carbon-14 in the fossilized tree branch would be approximately [tex]\( 1.02 \)[/tex] grams to three significant figures.
