I am not too sure if I did this question correct. Thanks in advanced!
Data table:
Time after Consumption (min): 30 60 90 120 150 180
Amount of Codeine in Blood (mg): 27.0 23.5 21.2 18.7 16.6 14.5
1.) create a scatter plot of the data and determine a suitable equation to model the amount of codeine in the bloodstream t min after taking the pill. Justify your choice of models.
This is what I did: I labelled the x-axis as Time (min) and the y-axis as Amount of Codeine (mg) and then I plotted each of the points.
After that I used the formula A=Pe^kt and got the equation A=31.0213e^-0.00463t. Would this equation be correct? Also, what does the question mean by "Justify your choice of models"
Justification: the amount in the blood at any time has to be related to the amount that was there previously, a tale-tale sign of an exponsntial decay.
Does your curve fit your scatter plot?
when I drew a curve through the scatter plot, i did get an exponential decay model. would that be all I would need for the justification part?
To determine whether your equation is correct and to provide a justification for your choice of models, we need to analyze the scatter plot and consider different modeling options.
First, plotting the given data points on a scatter plot is a good approach. By labeling the x-axis as "Time (min)" and the y-axis as "Amount of Codeine (mg)", you have correctly represented the two variables.
Regarding the equation A = Pe^kt, it is an exponential decay model commonly used to describe the decay of substances in the bloodstream over time. However, to determine if this model is suitable, we need to examine the scatter plot and check if it displays an exponential decay pattern.
After creating the scatter plot, observe the overall trend of the data points. If they gradually decrease in a smooth, curved manner, an exponential decay model would be a reasonable choice. If the points do not seem to follow an exponential decay pattern, alternative modeling options should be considered.
To calculate the values of P and k in the exponential decay model, known as the rate constant, we need to find specific points on the graph that correspond to the initial amount of codeine (P) and the decay constant (k). These values can be obtained using curve-fitting techniques or regression analysis.
Once you have the equation, A = Pe^kt, you can compare its predictions with the actual data points to determine if it is a good fit. One way to check this is by calculating the residuals, which are the differences between the predicted values from the equation and the actual data points. If the residuals are small and the predicted values closely match the observed data, the equation is likely a good fit.
The question asks you to justify your choice of models. Justification here means providing a rationale for selecting the exponential decay model. You can justify your choice by explaining how the scatter plot displayed a clear decay pattern, and how an exponential decay model is commonly used to describe the decrease in codeine concentration over time in the bloodstream. Additionally, you could highlight any assumptions made, such as assuming a constant decay rate, and discuss the practicality of the exponential decay model in this specific context.