Rory is staying in a cabin on a hill 300 feet above sea level. She walks down the hill to the water's edge. The equation of her average change in elevation over time is [tex]\(e = 300 - 10t\)[/tex], where [tex]\(t\)[/tex] is the time in minutes since she left the cabin, and [tex]\(e\)[/tex] is her elevation with regard to sea level.

Determine the viable points and their values in the table relating [tex]\(t\)[/tex] and [tex]\(e\)[/tex]:

[tex]\[
\begin{tabular}{|c|c|}
\hline
Time (\(t\)) & Elevation (\(e\)) \\
\hline
-2 & \(a\) \\
\hline
3.5 & \(b\) \\
\hline
30 & \(c\) \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{array}{l}
a = \square \checkmark \\
b = \square \checkmark \\
c = \square
\end{array}
\][/tex]



Answer :

To solve this problem, we need to evaluate the elevation [tex]\(e\)[/tex] using the given equation [tex]\(e = 300 - 10t\)[/tex] for the specified time points [tex]\(t = -2\)[/tex], [tex]\(t = 3.5\)[/tex], and [tex]\(t = 30\)[/tex].

1. Calculate [tex]\( e \)[/tex] when [tex]\( t = -2 \)[/tex]:
- Substitute [tex]\( t = -2 \)[/tex] into the equation [tex]\(e = 300 - 10t\)[/tex]:
[tex]\[ e = 300 - 10(-2) \][/tex]
- Simplify the expression:
[tex]\[ e = 300 + 20 \][/tex]
- Thus, the elevation [tex]\(e\)[/tex] when [tex]\( t = -2 \)[/tex] is:
[tex]\[ e = 320 \][/tex]

2. Calculate [tex]\( e \)[/tex] when [tex]\( t = 3.5 \)[/tex]:
- Substitute [tex]\( t = 3.5 \)[/tex] into the equation [tex]\(e = 300 - 10t\)[/tex]:
[tex]\[ e = 300 - 10(3.5) \][/tex]
- Simplify the expression:
[tex]\[ e = 300 - 35 \][/tex]
- Thus, the elevation [tex]\(e\)[/tex] when [tex]\( t = 3.5 \)[/tex] is:
[tex]\[ e = 265.0 \][/tex]

3. Calculate [tex]\( e \)[/tex] when [tex]\( t = 30 \)[/tex]:
- Substitute [tex]\( t = 30 \)[/tex] into the equation [tex]\(e = 300 - 10t\)[/tex]:
[tex]\[ e = 300 - 10(30) \][/tex]
- Simplify the expression:
[tex]\[ e = 300 - 300 \][/tex]
- Thus, the elevation [tex]\(e\)[/tex] when [tex]\( t = 30 \)[/tex] is:
[tex]\[ e = 0 \][/tex]

Now, we populate the table with the calculated values:

[tex]\[ \begin{array}{|c|c|} \hline \text{Time } (t) & \text{Elevation } (e) \\ \hline -2 & 320 \\ \hline 3.5 & 265.0 \\ \hline 30 & 0 \\ \hline \end{array} \][/tex]

Thus, the values in the table are:
[tex]\[ a = 320 \checkmark \\ b = 265.0 \checkmark \\ c = 0 \][/tex]