# IB Mathematics Portfolio - Modeling the amount of a drug in the bloodstream

MINZY NGUYEN

IB SL I

4/19/07

Modeling the amount of a drug in the bloodstream

The graph below records the amt of a drug for treating malaria in the bloodstream over 10 hrs following an initial dose of 10 micrograms.  Student must find a suitable function to model the data given and compare and analyze the student’s model to the given data.  Finally, interpret the data by stating limitation and modification for the model and how the model can be applied to real life situations.

Original Data Given:

Part A

Find a suitable function for your graph and comment on it.

MODEL A

MODEL A

A1.  This model demonstrates the exponential decay relationship between the time in hours and the amount of drug left in the bloodstream.  According to the model data, the amount of hours increases as the amount of drug in the blood stream decrease.  The function of the trend line is y= 10.391e^-1863x.  The coefficient of determination is .9942.  “The rate of decrease of the drug is approximately proportional to the amount remaining.”

A2.  Method

First, I start by making a scatter plot using the Excel Program.  I enter the x and y values (data) given on the task sheet into two columns. Then I add the exponential trend line for the line of best fit by right clicking the data points on the scatter plot and “add trend line”.  I chose the exponential trend line because after looking at the other available trend lines in the Excel program, the line that will provide me with the r2 value closest to 1 is the exponential decay trend line with the r2 value of .9942.  A trend line is most reliable and reasonable when the r2 value is at one or very close to one.  Also, the exponential line looks like it is the most reasonable and best fitting.  The graph I made is very similar to the model given.  The only difference is the original data didn’t have the line of best fit on it.  It still contains the same data points and scale.

The r2 or the coefficient of determination displays how closely my model fits the data. The coefficient of determination is “the ratio of the explained variation to the total variation. The coefficient of determination represents the percent of the data that is the closest to the line of best fit.”   “The coefficient of correlation of a set of pairs of quantities is equal to the sum of the products of the deviation of each quantity in the pairs from its respective mean, divided by the product of the number in the set and the standard deviations. It indicates the strength and direction of a linear relationship between two random variables.  The coefficient of determination is a measure of how well the regression line represents the data.  If the regression line passes exactly through every point on the scatter plot, it would be able to explain all of the variation. The further the line is away from the points, the less it is able to explain.”(Math bits)

We can use the following formula to calculate r2:

Explained Variance

True Variance                OR

The closer the value it is to 1.00, the more suitable the line is for the graph.  I just used the Excel program to calculate the r2 value for me.

A3.  Interpretation

The reasonableness of the data is acceptable for the most part with minor discrepancy.  The data is pretty reasonable until it goes beyond 10 or more hours when the amount of ...