The number N of operating drive-in movie theatres in the...

The number $N$ of operating drive-in movie theatres in the state of Virginia in various years $y$ is given in the table below. $$ \begin{array}{|r|r|r|r|r|r|r|} \hline y & 1958 & 1967 & 1972 & 1977 & 1982 & 1999 \\ \hline N & 143 & 90 & 102 & 87 & 56 & 9 \\ \hline \end{array} $$ (a) Find the average rate of change in the number of drive-in movie theatres in Virginia over each time interval between table entries. (b) Describe the units and real-world significance of these average rates of change. (c) Over which time period was the average rate of change the most drastic? On average, assuming that no new theatres were built, how many drive-ins closed per year during that period?

The number $N$ of operating drive-in movie theatres in the state of Virginia in various years $y$ is given in the table below.
$$
\begin{array}{|r|r|r|r|r|r|r|}
\hline y & 1958 & 1967 & 1972 & 1977 & 1982 & 1999 \\
\hline N & 143 & 90 & 102 & 87 & 56 & 9 \\
\hline
\end{array}
$$
(a) Find the average rate of change in the number of drive-in movie theatres in Virginia over each time interval between table entries.
(b) Describe the units and real-world significance of these average rates of change.
(c) Over which time period was the average rate of change the most drastic? On average, assuming that no new theatres were built, how many drive-ins closed per year during that period?

Key Concepts

Average Rate of Change

This concept describes how a quantity changes on average over a specific time span. It is computed by taking the difference in the function values at two points and dividing by the change in the independent variable, which in this context is time. This value provides an overall measure of the speed at which the quantity (such as the number of drive-in theatres) increases or decreases over an interval.

Interpretation of Slope in a Real-World Context

In practical applications, the average rate of change is equivalent to the slope of a line connecting two data points on a graph. It indicates, for example, how many theatres closed (or opened) per year between two years. This helps in understanding the trend and the intensity of change over different intervals.

Units of Measurement and Their Significance

The units of the average rate of change are determined by the units of the dependent and independent variables. In this case, since the quantity is the number of theatres and the time is given in years, the average rate of change is measured in theatres per year. This unit helps quantify the rate at which the theatres are either closing or emerging over time.

Comparative Analysis of Rates

This concept involves comparing the calculated average rates of change over different time intervals to identify which period experienced the most significant change. The interval with the greatest magnitude of the rate (whether positive or negative) is considered the period with the most drastic change. This analysis aids in pinpointing periods of rapid decline or growth within the dataset.

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