Answer:
2.1 times 10 to the power of 9 years
Explanation:
U-238 is a radioactive isotope that emits particles as it decays. This results in a reduction of its mass, converting it into Pb-206.
The duration required for a substance to lose half of its mass is defined as its half-life. By knowing both the initial mass (mi) and the resulting mass (m), the number of half-lives that have occurred (n) can be calculated using the following equation:
m = mi divided by 2 raised to the power of n
The mass of Pb-206 corresponds to the mass that U-238 has lost, thus it can be expressed as mi - m. Consequently, the mass ratio can be represented as:
(mi - m) divided by m = 0.337 divided by 1
mi - m = 0.337m
mi = 1.337m
Inserting mi into the half-life equation gives:
m = 1.337m divided by 2 raised to the power of n
2 raised to the power of n = 1.337m divided by m
2 raised to the power of n = 1.337
ln(2 raised to the power of n) = ln(1.337)
n multiplied by ln(2) = ln(1.337)
n = ln(1.337) divided by ln(2)
n = 0.4190
The elapsed time (t), or the approximate age of the sample, is calculated by multiplying the half-life duration by n:
t = 4.5 times 10 to the power of 9 multiplied by 0.4190
t ≈ 1.88 times 10 to the power of 9 years, which is approximately 2.1 times 10 to the power of 9 years.
