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Answer :
The amount left after 20 years will be approximately 217.28 grams.
a. Exponential decay model for the given radioactive substance is: \[A(t) = A_0e^{-kt}\]
We are given that the half-life of a radioactive substance is 38.2 years. This means that after every 38.2 years, the quantity of the substance gets halved, that is:\[A_0/2 = A_0e^{-38.2k}\]
Therefore,\[\frac{1}{2} = e^{-38.2k}\]\[\Rightarrow \ln\left(\frac{1}{2}\right) = -38.2k\]\[\Rightarrow k = \frac{\ln(2)}{38.2}\]
Substituting this value of k in the expression of A(t), we get the required exponential decay model for the given substance as: \[A(t) = A_0e^{-0.0182t}\]b.
We are given that a sample of 500 grams will decay to 400 grams. We need to find how long it will take. We can use the exponential decay model to solve this.
Substituting the given values, we get:\[400 = 500e^{-0.0182t}\]\[\Rightarrow e^{-0.0182t} = \frac{4}{5}\]\[\Rightarrow -0.0182t = \ln\left(\frac{4}{5}\right)\]\[\Rightarrow t = \frac{\ln\left(\frac{4}{5}\right)}{-0.0182}\]
Therefore, the sample will take about 8.9 years to decay from 500 grams to 400 grams.
c. We need to find how much of the substance will be left after 20 years, starting with an initial quantity of 500 grams. Using the given exponential decay model,\[A(20) = 500e^{-0.0182(20)}\]\[= 500e^{-0.364}\]\[= 217.28\]
Therefore, the amount left after 20 years will be approximately 217.28 grams.
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