To determine which equation represents the distance [tex]\( y \)[/tex] from the lighthouse based on the number of hours [tex]\( x \)[/tex], we can follow these steps:
1. Gather the Given Data Points:
We have the following table representing the distance from the lighthouse over time:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Number of Hours} & \text{Distance from Lighthouse (in oceanic miles)} \\
\hline
2 & 53 \\
\hline
4 & 95.5 \\
\hline
6 & 138 \\
\hline
8 & 180.5 \\
\hline
10 & 223 \\
\hline
12 & 265.5 \\
\hline
14 & 308 \\
\hline
16 & 350.5 \\
\hline
\end{array}
\][/tex]
2. Understand Linear Relationship:
Since the cruise ship is traveling at a uniform speed, the relationship between the distance [tex]\( y \)[/tex] and the time [tex]\( x \)[/tex] should be linear:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope (rate of speed) and [tex]\( b \)[/tex] is the y-intercept (initial distance).
3. Calculate Slope and Intercept:
First, we identify the values for [tex]\( x \)[/tex] (time in hours) and [tex]\( y \)[/tex] (distance from the lighthouse). We then use these values to determine the best-fit line.
4. Match with Given Options:
We compare the calculated slope and intercept with the given options:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Option} & \text{Equation} \\
\hline
\text{A} & y = 42.5x + 10.5 \\
\hline
\text{B} & y = 10.5x + 32 \\
\hline
\text{C} & y = 21.25x + 10.5 \\
\hline
\text{D} & y = 12.25x + 28.5 \\
\hline
\text{E} & y = 12.5x + 10.5 \\
\hline
\end{array}
\][/tex]
5. Select the Correct Option:
After comparing the options with the calculated slope and intercept, the correct equation that matches the data points is:
[tex]\[
\boxed{y = 21.25x + 10.5}
\][/tex]
Thus, the equation that best represents the distance from the lighthouse over time is given by option [tex]\( C \)[/tex].
