Problem 62 - Calculus for Biology and Medicine

Question

A radioactive isotope decays over time, following an exponential decay law. That is, the amount of isotope left at time \(t\) is predicted to be: $$W(t)=W_{0} e^{-\lambda t}$$ where \(W_{0}\) and \(\lambda\) are both coefficients. You measure the following data on the amount of isotope left in a particular sample, \(W\), at different times \(t .\) $$\begin{array}{lcccccc} \hline \boldsymbol{t} & 0 & 0.1 & 0.2 & 0.4 & 0.8 & 1.0 \\ \boldsymbol{W} & 113.2 & 63.7 & 66.0 & 32.1 & 13.1 & 3.89 \\ \hline \end{array}$$ (a) Use a least squares method to estimate the coefficients \(W_{0}\) and \(\lambda\). (b) When the fitted coefficients \(W_{0}\) and \(\lambda\) are input into the model, what is the predicted half-life of the isotope (that is, the time taken for the amount of isotope present to decay from \(W_{0}\) to \(\frac{1}{2} W_{0}\) )?

Step-by-Step Solution

Answer

Use the least squares method to find \( W_0 \) and \( \lambda \). The half-life \( t_{1/2} \) is \( \frac{\ln(2)}{\lambda} \).

Step 1: Logarithmic Transformation

To apply the least squares method, take the natural logarithm of both sides of the exponential decay equation to linearize it: \[ \ln(W(t)) = \ln(W_0) - \lambda t. \] This transforms the model into a linear form, where \( \ln(W(t)) \) is the dependent variable and \( t \) is the independent variable.

Step 2: Prepare Data for Linear Regression

Compute the natural log of each given \( W \) value: - \( \ln(113.2), \ln(63.7), \ln(66.0), \ln(32.1), \ln(13.1), \ln(3.89) \).This will become the new dataset to perform linear regression on, with the \( t \) values as the independent variable.

Step 3: Perform Linear Regression

Use linear regression on the transformed data to determine the coefficients \( a \) and \( b \) in the equation \( \ln(W(t)) = a - b t \). These correspond to \( a = \ln(W_0) \) and \( b = \lambda \). Calculate the slope \( b \) and the intercept \( a \) using the least squares formulas.

Step 4: Estimate Coefficients

Using your calculations from the regression, solve for \( W_0 \) and \( \lambda \). Since \( a = \ln(W_0) \), exponentiate \( a \) to find \( W_0 \), and take \( b \) directly as \( \lambda \).

Step 5: Compute Half-life from c

The half-life \( t_{1/2} \) can be found using the formula: \[ t_{1/2} = \frac{\ln(2)}{\lambda}. \]Use the estimated value of \( \lambda \) to compute the half-life.

Key Concepts

Radioactive Decay

Radioactive decay is a process where an unstable atomic nucleus loses energy by emitting radiation. This process reduces the amount of the original radioactive substance over time, which is known as exponential decay. The formula governing this process is given by \[ W(t) = W_0 e^{-\lambda t} \]where:

• \( W(t) \) is the amount of substance left at time \( t \),
• \( W_0 \) is the initial amount of the substance,
• \( \lambda \) is the decay constant, representing the rate of decay.

As the decay process continues, the quantity \( W(t) \) diminishes; the greater the value of \( \lambda \), the faster the decay occurs. This exponential relationship helps predict how much of a substance will remain after a certain time period.

Least Squares Method

The least squares method is a popular technique in statistics to find the best-fitting curve or line to a set of data points. It's used to minimize the differences between observed and predicted values. To apply the least squares method in radioactive decay, we start by transforming the exponential decay equation into a linear form. Using a logarithmic transformation, \[ \ln(W(t)) = \ln(W_0) - \lambda t \]This rearrangement allows us to perform linear regression because it converts the problem into a linear equation. The primary goal here is to identify the coefficients \( \ln(W_0) \) and \( \lambda \) by finding the line that best fits the transformed data.Linear regression is then applied to the transformed dataset, helping us determine the slope \( b \), which equals \( \lambda \), and the intercept \( a \), corresponding to \( \ln(W_0) \). This method ensures we have the most accurate estimates for these coefficients, reducing the residual sum of squares.

Half-life

Half-life is a key concept in understanding radioactive decay as it indicates the time required for half of the original quantity of a radioactive substance to decay. It's given by the equation:\[ t_{1/2} = \frac{\ln(2)}{\lambda} \]Here, \( t_{1/2} \) is the half-life, and \( \lambda \) is the previously mentioned decay constant.This concept is crucial because it gives us a practical measurement to assess the rate of decay without knowing the exact path over time. The shorter the half-life, the quicker a substance will decay to half of its initial amount. Understanding half-life allows for predictions regarding the behavior of radioactive materials in scientific and practical applications.

Linear Regression

Linear regression is an essential statistical tool for modeling the relationship between variables. In the context of exponential decay, once we have made the logarithmic transformation to the decay equation, linear regression is used to estimate the parameters.The linearized form of our decay equation is:\[ \ln(W(t)) = a - b t \]where \( a = \ln(W_0) \) and \( b = \lambda \). Using the least squares method, linear regression provides values for the slope \( b \) and intercept \( a \), minimizing the sum of squared differences between observed and predicted \( \ln(W(t)) \) values.By finding the best fit line through the spread of data points, linear regression allows us to make accurate predictions of the decay process. This statistical method proves invaluable for translating raw data into meaningful insights on decay patterns, essentially linking rates of radioactive decay with observable half-life calculations.