Answer: The the fossil is approximately 8,267 years old.
Step-by-step explanation: To solve this problem, we can use the formula for exponential decay:
N(t) = N₀ * (1/2)^(t/T)
where:
N(t) = the amount of carbon-14 remaining after time t
N₀ = the initial amount of carbon-14
T = the half-life of carbon-14
Let's assume that the fossilized leaf originally contained 100% of its normal amount of carbon-14. Since it now contains only 33% of its normal amount, we can say that:
N(t) = 0.33N₀
Substituting this into the formula above, we get:
0.33N₀ = N₀ * (1/2)^(t/5600)
Dividing both sides by N₀ and taking the logarithm base 2 of both sides, we get:
t/5600 = log₂(0.33)
t = 5600 * log₂(0.33)
t ≈ -14187 years
This result is negative, which doesn't make sense in the context of the problem. It means that the fossil must be older than our initial assumption of 100% carbon-14. Let's assume instead that the fossil originally contained 50% of its normal amount of carbon-14 (which is more realistic). Then, we can say that:
N(t) = 0.5N₀
Substituting this into the formula above, we get:
0.5N₀ = N₀ * (1/2)^(t/5600)
Dividing both sides by N₀ and taking the logarithm base 2 of both sides, we get:
t/5600 = log₂(0.5)
t = 5600 * log₂(0.5)
t ≈ 8267 years
Therefore, the fossil is approximately 8267 years old.