Let's break down the given problem step-by-step to determine the values for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
We are given the linear equation for Rory's elevation over time:
[tex]\[ e = 300 - 10t \][/tex]
where:
- [tex]\(e\)[/tex] is the elevation in feet.
- [tex]\(t\)[/tex] is the time in minutes since Rory left the cabin.
Let's substitute each given time value into the equation to find the corresponding elevations.
1. For [tex]\(t = -2\)[/tex]:
[tex]\[ e = 300 - 10(-2) \][/tex]
Substitute [tex]\(-2\)[/tex] for [tex]\(t\)[/tex]:
[tex]\[ e = 300 + 20 \][/tex]
[tex]\[ e = 320 \][/tex]
So, [tex]\(a = 320\)[/tex].
2. For [tex]\(t = 3.5\)[/tex]:
[tex]\[ e = 300 - 10(3.5) \][/tex]
Substitute [tex]\(3.5\)[/tex] for [tex]\(t\)[/tex]:
[tex]\[ e = 300 - 35 \][/tex]
[tex]\[ e = 265 \][/tex]
So, [tex]\(b = 265\)[/tex].
3. For [tex]\(t = 30\)[/tex]:
[tex]\[ e = 300 - 10(30) \][/tex]
Substitute [tex]\(30\)[/tex] for [tex]\(t\)[/tex]:
[tex]\[ e = 300 - 300 \][/tex]
[tex]\[ e = 0 \][/tex]
So, [tex]\(c = 0\)[/tex].
Thus, the values in the table relating [tex]\(t\)[/tex] and [tex]\(e\)[/tex] are:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Time }(t) & \text{Elevation }(e) \\
\hline
-2 & 320 \\
\hline
3.5 & 265 \\
\hline
30 & 0 \\
\hline
\end{array}
\][/tex]
So, the viable points and their values are:
[tex]\[
\begin{array}{l}
a = 320 \checkmark \\
b = 265 \\
c = 0 \\
\end{array}
\][/tex]
