Question

data were collected from libraries about the number of patrons who have overdue books and how long they have been using the library. the equation $hat{y}=9.5(0.8)^x$ is a model that fits the data. which of the following uses the model correctly and is an example of extrapolation? a patron who has been using the library for 12 years is predicted to have between 3 and 4 overdue books. a patron who has been using the library for 12 years is predicted to have less than 1 overdue book. a patron who has been using the library for 2 years is predicted to have between 5 and 6 overdue books. a patron who has been using the library for 2 years is predicted to have more than 6 overdue books.

Explanation:

Step1: Recall the concept of extrapolation

Extrapolation is using a model to make predictions outside the range of the known - data. The data ranges from 1 to 10 years of library - using experience.

Step2: Analyze each option

For a patron using the library for 12 years, we are going beyond the range of the data (1 - 10 years) in the table. We use the model $\hat{y}=9.5(0.8)^x$. When $x = 12$, $\hat{y}=9.5\times(0.8)^{12}$. Calculate $(0.8)^{12}\approx0.068719476736$ and $\hat{y}=9.5\times0.068719476736\approx0.653$. This is an example of extrapolation as we are predicting for a value of $x$ (12 years) outside the range of the data used to create the model.

Answer:

A patron who has been using the library for 12 years is predicted to have less than 1 overdue book.