As age increases, so does the likelihood of coronary heart...

As age increases, so does the likelihood of coronary heart disease (CHD). The percentage $P$ of people $x$ years old with signs of CHD is shown in the table. $$\begin{array}{rrrrrrrr}x & 15 & 25 & 35 & 45 & 55 & 65 & 75 \\ \hline P(\%) & 2 & 7 & 19 & 43 & 68 & 82 & 87\end{array}$$ (a) Evaluate $P(25)$ and interpret the answer. (b) Find a function that models the data. (c) Graph $P$ and the data. (d) At what age does a person have a $50 \%$ chance of having signs of CHD?

As age increases, so does the likelihood of coronary heart disease (CHD). The percentage $P$ of people $x$ years old with signs of CHD is shown in the table.
$$\begin{array}{rrrrrrrr}x & 15 & 25 & 35 & 45 & 55 & 65 & 75 \\ \hline P(\%) & 2 & 7 & 19 & 43 & 68 & 82 & 87\end{array}$$
(a) Evaluate $P(25)$ and interpret the answer.
(b) Find a function that models the data.
(c) Graph $P$ and the data.
(d) At what age does a person have a $50 \%$ chance of having signs of CHD?

Key Concepts

Solving Equations for Threshold Values

This involves setting the function equal to a specific target value and solving for the variable. This technique is critical when determining the point at which a certain condition is met, such as the threshold at which a probability or percentage reaches a given level, providing actionable insight into the modeled phenomenon.

Graphing Functions and Data

Graphing allows for visual comparison between the theoretical model and the actual data points. By plotting both on the same coordinate system, one can assess the accuracy of the model and understand trends or discrepancies, which is a key step in data analysis.

Interpretation of Function Outputs

Interpreting the output of a function means explaining the significance of the calculated value in the context of a real-world situation. This involves relating numerical results to physical or conceptual quantities, thereby providing meaningful insights based on the model.

Function Evaluation

This concept involves substituting a specific input into a given function to determine its corresponding output. It is essential in understanding how a function behaves at particular points and forms the basis for interpreting data in mathematical and applied contexts.

Function Modeling

Function modeling is the process of finding a mathematical function that fits a set of data points. This technique involves identifying a pattern in the data and selecting an appropriate formula, which can then be used to predict or explain behavior within the system. Common models include linear, exponential, logistic, or other types of functions based on the data's nature.

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