The half-life of radioactive strontium-90 is approximately 32 years. In 1961, radioactive strontium-90 was released into the atmosphere during testing of nuclear weapons, and was absorbed into people's bones. How many years does it take until only 8 percent of the original amount absorbed remains?
ouch. How did I read 8% as 1/8?
As usual, go with Reiny's answer!
To determine how many years it takes until only 8 percent of the original amount of radioactive strontium-90 absorbed remains, we can use the concept of half-life.
The half-life of a radioactive substance is the time it takes for half of the initial quantity to decay. In this case, the half-life of strontium-90 is given as approximately 32 years.
To find out how many half-lives it takes until only 8 percent (0.08) of the original amount remains, we can use the following formula:
N = N₀ * (1/2)^(t / T₁/₂)
Where:
N = the final amount remaining (0.08 of the original)
N₀ = the initial amount (1.00 or 100%)
t = the number of years that have passed
T₁/₂ = the half-life of the substance (32 years)
Now, let's rearrange the formula to solve for t:
t = T₁/₂ * log₂(N / N₀)
Substituting the given values:
t = 32 * log₂(0.08 / 1.00)
Using a scientific calculator or computer, evaluate the expression inside the logarithm:
t ≈ 32 * log₂(0.08) ≈ 32 * (-3.3219) ≈ -106.14
The result is approximately -106.14 years. However, this negative value does not make sense in our context because it represents a time before 1961. Therefore, we can ignore the negative sign and consider only the absolute value, giving us approximately 106.14 years.
Hence, it would take approximately 106.14 years until only 8 percent of the original amount of radioactive strontium-90 absorbed remains.
Since 1/8 = (1/2)^3, it will take 3 half-lives, or 96 years.
I would say
.08 = (1/2)^(t/32)
ln .08 = (t/32) ln .5
t/32 = ln .08/ln .5 = 3.6438..
t = 116.6 years
rough check:
after 32 years --- 50% left
after 64 years --- 25% left
after 96 years --- 12.5% left , (which is 1/8 but not 8%)
after about 16 more years we would lose about 1/4 of what was left, leaving appr 3/4 of 12.5 or appr 9%
answer is quite plausible