Answer:
To calculate the minimum age of the Earth, we can use the fact that the amount of uranium-235 remaining after a certain number of half-lives is given by the equation:
N = N0 * (1/2)^n
where N is the current amount of uranium-235, N0 is the initial amount of uranium-235, and n is the number of half-lives that have passed.
If we assume that the initial amount of uranium-235 in the oldest rocks on Earth was N0, and that 1/64th of the expected amount is present, then the current amount of uranium-235 is:
N = N0 * (1/64)
We can set this equal to the amount of uranium-235 that would remain after n half-lives:
N = N0 * (1/2)^n
N0 * (1/64) = N0 * (1/2)^n
Dividing both sides by N0 gives:
1/64 = (1/2)^n
To solve for n, we can take the logarithm of both sides of the equation:
log(1/64) = log((1/2)^n)
Using the logarithmic property that log(a^b) = b*log(a), we can simplify the right side of the equation:
log(1/64) = n*log(1/2)
Dividing both sides by log(1/2) gives:
n = log(1/64) / log(1/2)
n ≈ 6
Therefore, 6 half-lives have passed since the initial amount of uranium-235 was present in the oldest rocks on Earth.
The age of the Earth can be calculated by multiplying the number of half-lives by the half-life time:
Age of the Earth = n * half-life time
Age of the Earth = 6 * 704 million years
Age of the Earth ≈ 4.2 billion years
Therefore, the minimum age of the Earth is approximately 4.2 billion years.