To determine the correct equation that represents the depth of the lake [tex]\( y \)[/tex] based on the number of weeks passed [tex]\( x \)[/tex], we will perform a linear regression analysis to derive the equation of the line that best fits the given data points.
### Step-by-Step Analysis:
1. Initial Setup:
We have the following pairs of data points [tex]$(x, y)$[/tex] where [tex]\( x \)[/tex] is the number of weeks and [tex]\( y \)[/tex] is the lake depth in feet:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Time (weeks)} & \text{Lake Depth (feet)} \\
\hline
0 & 346.0 \\
\hline
2 & 344.8 \\
\hline
5 & 343.0 \\
\hline
7 & 341.8 \\
\hline
10 & 340.0 \\
\hline
12 & 338.8 \\
\hline
\end{array}
\][/tex]
2. Determine the Equation of the Line:
- First, we need to find the slope ([tex]\( m \)[/tex]) and the y-intercept ([tex]\( b \)[/tex]) of the line in the form [tex]\( y = mx + b \)[/tex].
3. Calculate the Slope:
- The slope [tex]\( m \)[/tex] can be calculated using the formula:
[tex]\[
m = \frac{\Delta y}{\Delta x}
\][/tex]
From the given data, the depth decreases steadily. To find [tex]\( \Delta y \)[/tex] (change in depth) and [tex]\( \Delta x \)[/tex] (change in weeks), let’s consider the change from 0 weeks to 2 weeks:
[tex]\[
m = \frac{344.8 - 346.0}{2 - 0} = \frac{-1.2}{2} = -0.6
\][/tex]
4. Determine the y-Intercept:
- The y-intercept [tex]\( b \)[/tex] is the depth of the lake when [tex]\( x = 0 \)[/tex]:
[tex]\[
b = 346
\][/tex]
5. Construct the Equation:
- Using the slope and y-intercept calculated:
[tex]\[
y = -0.6x + 346
\][/tex]
### Conclusion:
The equation that best represents the relationship between the lake's depth [tex]\( y \)[/tex] and the number of weeks [tex]\( x \)[/tex] is:
[tex]\[
\boxed{y = -0.6x + 346}
\][/tex]
Thus, the correct answer is option A.
