About half of a sample of K-40 will decay into Ar-40 in 1 billion years. Suppose we find a rock sample that has 12% K-40. Approximately how old must it be?

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To tackle this problem, we start from the given hint: about half of K-40 decays in roughly 1 billion years, so we can treat the half-life as ~1 x 10^9 years.
Option 1: 3 billion years old. If the half-life is ~1 billion years, after 3 half-lives the remaining fraction is (1/2)^3 = 1/8 = 12.5%, which is very close to the given 12% remaining. This aligns with the exponential decay model and the rough half-life provided.
Option 2: 2 billion years old. After two half-lives, the remaining fraction would be (1/2)^2 = 1/4 = 25%. That is far larger than 12%, so this would correspond to a younger age than the rock is stated to be.
Option 3: 1 billion years old. After one half-life, the remaining fraction would be (1/2) = 50%. This is not near the 12% given, so it underestimates the age.
Option 4: 500 million years old. After half a half-life (0.5 x 1 billion years), the remaining fraction would be roughly (1/2)^{0.5} ≈ 0.707, i.e., about 70.7% remains. This is far from 12%, indicating a much younger age than the rock would be.
Overall, the 12% remaining K-40 corresponds to about three half-lives elapsed, which is around 3 billion years given the stated half-life time. The other options yield remaining fractions that are inconsistent with 12% remaining when using the same decay rule.

ASTR-1-45046-F25 Lecture: Solar System Formation

About half of a sample of K-40 will decay into Ar-40 in 1 billion years. Suppose we find a rock sample that has 12% K-40. Approximately how old must it be?

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