Box Office Receipts U.S. movie box office receipts, in...

Box Office Receipts U.S. movie box office receipts, in billions of dollars, are shown in 5-year increments from 1997 to 2017 . $$ \begin{array}{c|c} \text { Year } & \text { Receipts } \\ \hline 1997 & 6.366 \\ \hline 2002 & 9.155 \\ \hline 2007 & 9.664 \\ \hline 2012 & 10.838 \\ \hline 2017 & 11.072 \\ \hline \end{array} $$ These receipts $R$ are reasonably approximated by the linear model $$ R=0.2219 x+7.200 $$ where $x=0$ corresponds to $1997, x=5$ corresponds to $2002,$ and so on. Using the model, calculate the year in which the receipts first exceed each amount. (a) 8.1$ billion (b) 10.6$ billion

Box Office Receipts U.S. movie box office receipts, in billions of dollars, are shown in 5-year increments from 1997 to 2017 . $$
\begin{array}{c|c}
\text { Year } & \text { Receipts } \\
\hline 1997 & 6.366 \\
\hline 2002 & 9.155 \\
\hline 2007 & 9.664 \\
\hline 2012 & 10.838 \\
\hline 2017 & 11.072 \\
\hline
\end{array}
$$
These receipts $R$ are reasonably approximated by the linear model
$$
R=0.2219 x+7.200
$$
where $x=0$ corresponds to $1997, x=5$ corresponds to $2002,$ and so on. Using the model, calculate the year in which the receipts first exceed each amount.
(a) 8.1$ billion
(b) 10.6$ billion

Key Concepts

Extrapolation

Extrapolation is the process of using a linear model to estimate values outside the range of the observed data. It relies on the assumption that the established trend continues beyond the known data points and is useful for making predictions about future or past values.

Solving Linear Equations and Inequalities

Finding threshold values within a linear model typically involves setting the linear equation equal to or greater than a given target value and then solving for the independent variable. This process requires algebraic manipulation of the linear equation to determine when the dependent variable exceeds or meets the specified condition.

Interpreting the Independent Variable in Context

In many models, the independent variable is defined relative to a specific reference point, making it crucial to understand the mapping between the model variable and its real-world counterpart. This allows for the correct translation of the model's output back to the actual scenario, such as converting a time-based numerical value to a specific calendar year.

Linear Models

Linear models are mathematical representations that describe a relationship between two variables using a straight line. They are often used to approximate trends in data by relating an independent variable (such as time) to a dependent variable (such as a measured quantity) through a constant rate of change and a constant baseline, captured in the formula y = mx + b.

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