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 y = 12(0.81)^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 13 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 more than 94 overdue books. a patron who has been using the library for 13 years is predicted to have between 6 and 7 overdue books. a patron who has been using the library for 2 years is predicted to have between 7 and 8 overdue books. years using library number of overdue books 1 12 1.5 9 3 10 3.5 8 5 5 7 4 10 2

Explanation:

Step1: Recall the concept of extrapolation

Extrapolation uses a model to predict values outside the known - data range. The equation is $y = 12(0.81)^x$, where $y$ is the number of overdue books and $x$ is the number of years using the library.

Step2: Analyze each option

• For a patron using the library for 13 years: Substitute $x = 13$ into $y=12(0.81)^{13}$. Calculate $y = 12\times(0.81)^{13}\approx12\times0.044\approx0.53<1$.
• For a patron using the library for 2 years: Substitute $x = 2$ into $y = 12(0.81)^{2}=12\times0.6561 = 7.8732<9.4$.
• For a patron using the library for 13 years: We already saw the value is less than 1, not between 6 and 7.
• For a patron using the library for 2 years: The value is approximately 7.87, not between 7 and 8.

Answer:

A patron who has been using the library for 13 years is predicted to have less than 1 overdue book.