A drug is injected into a patient and the concentration of...

A drug is injected into a patient and the concentration of the drug in the bloodstream is monitored. The drug's concentration, $C(t)$, in milligrams per liter, after $t$ hours is modeled by $$ C(t)=\frac{5 t}{t^2+1} . $$ The graph of this rational function, obtained with a graphing utility, is shown in the figure. GRAPGH CANT COPY a. Use the graph to obtain a reasonable estimate of the drug's concentration after 3 hours. b. Use the function's equation displayed in the voice balloon to determine the drug's concentration after 3 hours. c. Use the function's equation to find the horizontal asymptote for the graph. Describe what this means about the drug's concentration in the patient's bloodstream as time increases. Among all deaths from a particular disease, the percentage that are smoking related (21-39 cigarettes per day) is a function of the disease's incidence ratio. The incidence ratio describes the number of times more likely smokers are than nonsmokers to die from the disease. The following table shows the incidence ratios for heart disease and lung cancer for two age groups. $$ \begin{array}{|l|c|c|} \hline {\text { Incidence Ratios }} \\ \hline & \text { Heart Disease } & \text { Lung Cancer } \\ \hline \text { Ages 55-64 } & 1.9 & 10 \\ \hline \text { Ages 65-74 } & 1.7 & 9 \\ \hline \end{array} $$

A drug is injected into a patient and the concentration of the drug in the bloodstream is monitored. The drug's concentration, $C(t)$, in milligrams per liter, after $t$ hours is modeled by
$$
C(t)=\frac{5 t}{t^2+1} .
$$
The graph of this rational function, obtained with a graphing utility, is shown in the figure.
GRAPGH CANT COPY
a. Use the graph to obtain a reasonable estimate of the drug's concentration after 3 hours.
b. Use the function's equation displayed in the voice balloon to determine the drug's concentration after 3 hours.
c. Use the function's equation to find the horizontal asymptote for the graph. Describe what this means about the drug's concentration in the patient's bloodstream as time increases.

Among all deaths from a particular disease, the percentage that are smoking related (21-39 cigarettes per day) is a function of the disease's incidence ratio. The incidence ratio describes the number of times more likely smokers are than nonsmokers to die from the disease. The following table shows the incidence ratios for heart disease and lung cancer for two age groups.
$$
\begin{array}{|l|c|c|}
\hline {\text { Incidence Ratios }} \\
\hline & \text { Heart Disease } & \text { Lung Cancer } \\
\hline \text { Ages 55-64 } & 1.9 & 10 \\
\hline \text { Ages 65-74 } & 1.7 & 9 \\
\hline
\end{array}
$$

Key Concepts

Incidence Ratio

The incidence ratio is a measure used in epidemiology to compare the likelihood of an event, such as death from a disease, between two groups. It quantifies how many times more likely one group (e.g., smokers) is to experience an event compared to another group (e.g., nonsmokers). This concept is crucial for understanding and analyzing risk factors and their impact on health outcomes.

Function Modeling

Function modeling is the process of using mathematical functions to represent real-world phenomena. It involves formulating a function that captures essential features of a system, such as growth, decay, and equilibrium states. This method allows for predictions about system behavior and is widely used in contexts like pharmacokinetics and epidemiology.

Graphical Analysis of Functions

Graphical analysis involves using visual representations of functions to estimate values and understand overall behavior. This approach is particularly useful when numerical methods or calculus are infeasible or as a way to check analytical results, providing an intuitive overview of how the function behaves over its domain.

Horizontal Asymptotes

A horizontal asymptote represents a line that the graph of a function approaches as the independent variable grows large in magnitude. In the context of a physical model, such as drug concentration in the bloodstream, the horizontal asymptote indicates the limiting behavior of the system—showing what value the concentration tends toward as time increases indefinitely.

Rational Functions

Rational functions are functions expressed as the ratio of two polynomials. They are important in modeling because they can capture behaviors such as peaks, decay, and limits that are common in real-world phenomena. Understanding their structure, including potential asymptotes and intercepts, helps in both the analysis and interpretation of the behavior of the system being modeled.

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