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Making a Table of Values for a Linear Equation

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

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][tex]$e=300-10t$[/tex][/tex], where [tex][tex]$t$[/tex][/tex] is the time in minutes since she left the cabin, and [tex][tex]$e$[/tex][/tex] is her elevation with regard to sea level.

Which values are viable points, and what are their values in the table relating [tex][tex]$t$[/tex][/tex] and [tex][tex]$e$[/tex][/tex]?

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



Answer :

GinnyAnswer
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]

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