`U^(238)` is found to be in secular equilibrium with `Ra^(226)` on its ore. If chemical analysis shows 1 nuclei of `Ra^(226)` per `3.6xx10^(-6)` nuclei of `U^(238)`, find the half-life of `U^(238)`. Given the half-life is 1500 years.

Explanation

To solve the problem, we need to apply the concept of secular equilibrium between the radioactive isotopes Uranium-238 (U-238) and Radium-226 (Ra-226). The relationship in secular equilibrium states that the decay rate of the parent isotope (U-238) equals the decay rate of the daughter isotope (Ra-226).

### Step-by-Step Solution:

1. **Understanding Secular Equilibrium**:
In secular equilibrium, the decay rates of the parent and daughter isotopes are equal:
\[
\lambda_U N_U = \lambda_{Ra} N_{Ra}
...