Answer:
160 years.
Step-by-step explanation:
From the question given above, the following data were obtained:
Initial count rate (Cᵢ) = 400 count/min
Half-life (t½) = 40 years
Final count rate (Cբ) = 25 count/min
Time (t) =?
Next, we shall determine the number of half-lives that has elapse. This can be obtained as follow:
Initial count rate (Cᵢ) = 400 count/min
Final count rate (Cբ) = 25 count/min
Number of half-lives (n) =?
Cբ = 1/2ⁿ × Cᵢ
25 = 1/2ⁿ × 400
Cross multiply
25 × 2ⁿ = 400
Divide both side by 25
2ⁿ = 400/25
2ⁿ = 16
Express 16 in index form with 2 as the base
2ⁿ = 2⁴
n = 4
Thus, 4 half-lives has elapsed.
Finally, we shall determine the time taken for the radioactive material to decay to the rate of 25 counts per minute. This can be obtained as follow:
Half-life (t½) = 40 years
Number of half-lives (n) = 4
Time (t) =?
n = t / t½
4 = t / 40
Cross multiply
t = 4 × 40
t = 160 years.
Thus, it will take 160 years for the radioactive material to decay to the rate of 25 counts per minute.