### Functions and graphs: Relations and functions

### Example 2: The Widmark model

A second example of a linear relationship between two variables is the Widmark model of blood alcohol concentration in the human body after alcohol consumption in relation to the time that has elapsed after drinking.

The Swedish physiologist Erik P. Widmark (1889-1945) published the following formula for the blood alcohol concentration (BAC) in 1932: \[\mathrm{BAC}=\frac{D}{r\cdot G}-\beta\cdot t,\] where \(D\) is the amount of alcohol consumed (in grams), \(G\) body weight (in kg), \(r\) is called the Widmark factor (in L/kg), \(\beta\) is the constant elimination rate (in grams per liter per hour), and \(t\) is the time (in hours) after consumption of the alcohol.

In the above example, there are more than two variables, but we still speak of a linear relationship between blood alcohol concentration and time. We have in fact secretly chosen time \(t\) as the **independent variable,** i.e., the variable the value of which can be freely chosen. The blood alcohol concentration BAC is called the **dependent variable,** because its value depends on the value of the independent variable. The other variables are **parameters**, which are in this case perosn-based constants. In this example, the relationship between BAC and \(t\) is a function and BAC is ** a function of** \(t\), denoted \(\displaystyle\mathrm{BAC}(t)=\frac{D}{r\cdot G}-\beta\cdot t.\) The right-hand side is also called the

**function definition**or

**function rule**.