Final answer:
To determine how long it will take for the strontium-90 concentration to drop from 10.0 ppm to 1.25 ppm, we apply the half-life formula for exponential decay. Calculations reveal it will take three half-lives, or 86.4 years, which does not match any of the given options.
Step-by-step explanation:
The question is asking how long it would take for the concentration of strontium-90 in a milk sample to decrease from 10.0 ppm to 1.25 ppm, given that strontium-90 has a half-life of 28.8 years. To answer this, we use the half-life formula for exponential decay:
N(t) = N_0 (1/2)^(t/T)
Where:
- N(t) is the future amount of substance after time t
- N_0 is the initial amount of substance
- t is the time elapsed
- T is the half-life of the substance
In this case, we have:
- N_0 = 10.0 ppm
- N(t) = 1.25 ppm
- T = 28.8 years
We want to find t, so we rearrange the formula to solve for t:
t = T * (log(N(t)/N_0) / log(1/2))
Substituting our values in:
t = 28.8 years * (log(1.25/10.0) / log(1/2))
Calculate the time:
t ≈ 28.8 years * (log(0.125) / log(0.5))
t ≈ 28.8 years * (log(1/8) / log(1/2))
t ≈ 28.8 years * (-3 / -1)
t ≈ 86.4 years
Thus, it will take approximately 86.4 years for the concentration to drop to 1.25 ppm. However, this answer does not match any of the options given, indicating there might be an error in the question or the options provided. In reality, the number of half-lives to go from 10 ppm to 1.25 ppm is three (since 10 -> 5 -> 2.5 -> 1.25), and thus the time would be three half-lives, or 3 * 28.8 years, which is 86.4 years.