While on a trip, you notice that the video screen on the airplane, in addition to showing movies and news, records your altitude (in kilometers) above the ground. As the plane starts its descent (at time [tex][tex]$x=0$[/tex][/tex]), you record the following data:

\begin{tabular}{|l|l|}
\hline Time, [tex][tex]$x$[/tex][/tex] (min.) & Altitude, [tex][tex]$y$[/tex][/tex] (km) \\
\hline 0 & 12 \\
\hline 2 & 10 \\
\hline 4 & 8 \\
\hline 6 & 6 \\
\hline 8 & 4 \\
\hline 10 & 2 \\
\hline
\end{tabular}

If you were to graph these points, you would notice that the line represents a linear function. What is the practical meaning of the slope in this situation?

A. For every minute that passes, the airplane descends 1 kilometer.
B. For every minute that passes, the airplane ascends 1 kilometer.
C. For every 2 minutes that pass, the airplane descends 1 kilometer.
D. For every minute that passes, the airplane ascends 2 kilometers.



Answer :

To determine the practical meaning of the slope in this situation, let us analyze the given data for the plane's descent:

[tex]\[ \begin{array}{|c|c|} \hline \text{Time, } x \text{ (min.)} & \text{Altitude, } y \text{ (km)} \\ \hline 0 & 12 \\ \hline 2 & 10 \\ \hline 4 & 8 \\ \hline 6 & 6 \\ \hline 8 & 4 \\ \hline 10 & 2 \\ \hline \end{array} \][/tex]

Step-by-step solution to find the slope:

1. Identify two points on the graph: Let's use the first two points [tex]\((0, 12)\)[/tex] and [tex]\((2, 10)\)[/tex].

2. Calculate the slope [tex]\( m \)[/tex] using the formula for the slope of a line through two points:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
where [tex]\((x_1, y_1) = (0, 12)\)[/tex] and [tex]\((x_2, y_2) = (2, 10)\)[/tex].

3. Plug in the values:
[tex]\[ m = \frac{10 - 12}{2 - 0} = \frac{-2}{2} = -1 \][/tex]

4. Interpret the slope:
- The slope [tex]\( m = -1 \)[/tex] means the altitude decreases by 1 kilometer for every 1 minute that passes. In other words, the plane descends.

Thus, the practical meaning of the slope is:
[tex]\[ \text{a. For every minute that passes the airplane descends 1 kilometer.} \][/tex]

Therefore, the correct answer is:
a. For every minute that passes the airplane descends 1 kilometer.

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