Final answer:
To find the time it takes for the decay rate to fall to a certain value, we need to use the concept of half-life. In this case, the half-life of Tl−201 is given as 3.042 days. By using the formula for the number of half-lives, we can calculate the time it takes for the decay rate to reach 297 /s.
Step-by-step explanation:
To find the time it takes for the decay rate to fall to 297 /s, we need to use the concept of half-life. The half-life of Tl−201 is given as 3.042 days, which is equivalent to 3.042 x 24 x 60 x 60 = 262,876.8 seconds.
The decay rate is halved every half-life, so we can set up the following equation to solve for the number of half-lives it takes to reach 297 /s: (5.88x10^4 /s) / (2^x) = 297 /s.
Solving this equation, we find that x = log2((5.88x10^4) / (297)), which gives us approximately 6.79 half-lives. Since each half-life is 262,876.8 seconds, the total time it takes for the decay rate to fall to 297 /s is 6.79 x 262,876.8 = 1,787,438.912 seconds, or approximately 20.68 days.