The isotope potassium-40 (40K) decays into the isotope argon-40 (40Ar) with a half-life of 1.25 billion years. Suppose we find a rock in which seven-eighths of the potassium-40 has decayed to become argon-40, and only one-eighth of the original potassium-40 remains.

1/8 = (1/2)^3

so, 3 half-lives have passed.

2.6 billion years

69 hahahahaha

69.420 billion years lol ()

To solve this problem, we need to determine the age of the rock based on the given information. Here's how we can do that:

1. Identify the fraction of potassium-40 that remains: We are told that one-eighth of the original potassium-40 remains. Since one-eighth is equivalent to 1/8 or 1 divided by 8, the fraction of remaining potassium-40 is 1/8.

2. Determine the fraction of potassium-40 that has decayed: The remaining fraction of potassium-40 is complementary to the fraction that has decayed. Since 7/8 of the potassium-40 has decayed, the fraction of potassium-40 that has decayed is (7/8).

3. Calculate the number of half-lives: Each half-life represents a reduction of the initial amount of potassium-40 by half. To find the number of half-lives that have occurred, we can use the formula:
Number of half-lives = (log(base 2) of (fraction of remaining potassium-40)) / (log(base 2) of (fraction of decayed potassium-40))

For this problem, the fraction of remaining potassium-40 is 1/8, and the fraction of decayed potassium-40 is 7/8. So we can plug these values into the formula:

Number of half-lives = (log₁₀(1/8)) / (log₁₀(7/8))

4. Calculate the age of the rock: The age of the rock can be determined by multiplying the number of half-lives by the half-life of the isotope. In this case, the half-life of potassium-40 is given as 1.25 billion years.

Age of the rock = (Number of half-lives) * (half-life of potassium-40)

By substituting the value of the number of half-lives into the formula:

Age of the rock = (Number of half-lives) * (1.25 billion years)

Follow these steps to calculate the age of the rock.