This exercise uses the radioactive decay model.

The half-life of cesium-137 is 30 years. Suppose we have a 13-gram sample.

AFind a function $m(t)=m_0 2^{-t / h}$ that models the mass remaining after $t$ years.

BFind a function $m(t)=m_0 e^{rt}$ that models the mass remaining after $t$ years. (Round your $r$ value to four decimal places.)

CHow much of the sample (in grams) will remain after 83 years? (Round your answer to one decimal place.)

DAfter how many years will only 2 g of the sample remain? (Round your answer to the nearest whole number.)

To model the radioactive decay of Cesium-137, we start with an initial mass $m_0 = 13$ grams and a half-life $h = 30$ years. The half-life is the time it takes for the substance to decay to half of its initial amount.

AThe general model for radioactive decay in the form $m(t) = m_0 2^{-t/b}$ relates the half-life $h$ to the parameter $b$. Since $m(h) = \frac{1}{2}m_0$, we have $m_0 2^{-h/b} = \frac{1}{2}m_0$, which simplifies to $2^{-h/b} = 2^{-1}$. Thus, $h/b = 1$, meaning $b = h = 30$. The function is $m(t) = 13 \cdot 2^{-t/30}$.

BWe look for a model in the form $m(t) = m_0 e^{rt}$. Using the relation established by the half-life, we equate the two forms: $13 \cdot e^{rt} = 13 \cdot 2^{-t/30}$. Taking the natural logarithm of both sides gives $rt = -t/30 \ln(2)$, which simplifies to $r = -\frac{\ln(2)}{30}$. Calculating this value, we find $r \approx -0.0231$. The model is $m(t) = 13 \cdot e^{-0.0231t}$.

CTo find the mass remaining after $83$ years, we evaluate the decay model at $t = 83$: $m(83) = 13 \cdot e^{-0.0231 \cdot 83} \approx 13 \cdot e^{-1.9173} \approx 1.91$ grams. Rounding to one decimal place, the remaining mass is $1.9$ grams.

DTo find when the remaining mass is $2$ grams, we set $m(t) = 2$ and solve for $t$: $2 = 13 \cdot e^{-0.0231 \cdot t}$. Dividing by $13$ gives $\frac{2}{13} = e^{-0.0231 \cdot t}$. Taking the natural logarithm, we have $\ln(\frac{2}{13}) = -0.0231 \cdot t$. Solving for $t$, we get $t = \frac{\ln(2/13)}{-0.0231} \approx 81.01$. Rounding to the nearest whole number, we find $t \approx 81$ years.

Examples

Radioactive decay models are essential in fields like archeology and medicine. For example, carbon-14 dating uses similar exponential decay equations to estimate the age of ancient organic materials. By measuring the remaining radioactive carbon in a sample and comparing it to the initial concentration, researchers can determine when the organism died, providing a precise timeline for historical artifacts and fossilized remains. These calculations help us understand the age of objects that are thousands of years old.