A sample of Tl-201 has an initial decay rate of 5.88⨉10⁴/s. How long will it take for the decay rate to fall to 287/s? (Tl-201 has a half-life of 3.042 days.)

Verified step by step guidance

1. Identify the initial decay rate \( R_0 = 5.88 \times 10^4 \text{ s}^{-1} \) and the final decay rate \( R = 287 \text{ s}^{-1} \).
2. Use the decay formula \( R = R_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \), where \( t_{1/2} \) is the half-life.
3. Substitute the known values into the decay formula: \( 287 = 5.88 \times 10^4 \times \left( \frac{1}{2} \right)^{\frac{t}{3.042}} \).
4. Solve for \( t \) by taking the natural logarithm of both sides to isolate \( t \).
5. Calculate \( t \) using the equation \( t = 3.042 \times \frac{\ln(287/5.88 \times 10^4)}{\ln(1/2)} \).