A sample of a radioactive isotope had an initial mass of 370 mg in the year 2000 and decays exponentially over time. A measurement in the year 2008 found that the sample's mass had decayed to 110 mg. What would be the expected mass of the sample in the year 2017, to the nearest whole number?
We can model the decay of the radioactive isotope using the equation:
m(t) = m₀ * e^(kt)
where:
m(t) is the mass of the isotope at time t,
m₀ is the initial mass of the isotope,
k is the decay constant, and
t is the time elapsed.
We are given that the initial mass, m₀, is 370 mg in the year 2000, and the mass, m(t), is 110 mg in the year 2008. Let's use this information to find the decay constant, k.
Substituting the given values into the equation, we have:
110 mg = 370 mg * e^(k * 8)
Dividing both sides of the equation by 370 mg:
110 mg / 370 mg = e^(k * 8)
0.297 = e^(8k)
To isolate the variable, k, we can take the natural logarithm (ln) of both sides of the equation:
ln(0.297) = ln(e^(8k))
ln(0.297) = 8k
Using a calculator, we find:
k ≈ -0.1331
Now, we can use the decay constant, k, to find the expected mass of the radioactive isotope in the year 2017, which is 17 years after the initial measurement in 2000.
Substituting the values into the equation, we have:
m(t) = 370 mg * e^(-0.1331 * 17)
Using a calculator, we find:
m(t) ≈ 113 mg
Therefore, the expected mass of the sample in the year 2017 would be approximately 113 mg.