Certainly! Let's solve the problem step-by-step given the equation [tex]\( e = 300 - 10t \)[/tex], where [tex]\( e \)[/tex] is the elevation in feet, and [tex]\( t \)[/tex] is the time in minutes.
### Time (t) = -2:
1. Substitute [tex]\( t = -2 \)[/tex] into the equation:
[tex]\[
e = 300 - 10(-2)
\][/tex]
2. Simplify the equation:
[tex]\[
e = 300 + 20 = 320
\][/tex]
3. So, [tex]\( a = 320 \checkmark \)[/tex] (indicating it is a viable point since Rory could have been at this elevation before starting her descent).
### Time (t) = 3.5:
1. Substitute [tex]\( t = 3.5 \)[/tex] into the equation:
[tex]\[
e = 300 - 10(3.5)
\][/tex]
2. Simplify the equation:
[tex]\[
e = 300 - 35 = 265.0
\][/tex]
3. So, [tex]\( b = 265.0 \checkmark \)[/tex] (indicating it is a viable point).
### Time (t) = 30:
1. Substitute [tex]\( t = 30 \)[/tex] into the equation:
[tex]\[
e = 300 - 10(30)
\][/tex]
2. Simplify the equation:
[tex]\[
e = 300 - 300 = 0
\][/tex]
3. So, [tex]\( c = 0 \)[/tex] (indicating this point is viable; Rory reaches sea level at this time).
### Summary:
After checking the points, we complete the table as follows:
\begin{center}
\begin{tabular}{|c|c|}
\hline
Time (t) & Elevation (e) \\
\hline
-2 & 320 \\
\hline
3.5 & 265.0 \\
\hline
30 & 0 \\
\hline
\end{tabular}
\end{center}
And in the summarized form:
[tex]\[
\begin{array}{l}
a = 320 \checkmark \\
b = 265.0 \checkmark \\
c = 0
\end{array}
\][/tex]
