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The table you provided shows the relationship between the depth (in feet) and the temperature (in °F). Based on this data, I will find an equation that models this relationship.

First, let's summarize the data points:

\[
\begin{array}{|c|c|}
\hline
\text{Depth (ft)} & \text{Temperature (°F)} \\
\hline
0 & 74 \\
10 & 72 \\
20 & 71 \\
30 & 61 \\
40 & 55 \\
50 & 53 \\
\hline
\end{array}
\]

This looks like a decreasing linear or piecewise linear relationship, but since there is a sharp drop between depths of 20 and 30 feet, a linear approximation may not be perfect. However, we can first try to fit a linear equation and see how it behaves.

The general form of a linear equation is:

\[
T(d) = m \cdot d + b
\]

Where:
- \( T(d) \) is the temperature at depth \( d \),
- \( m \) is the slope of the line,
- \( b \) is the y-intercept (the temperature when \( d = 0 \)).

To determine \( m \) (the slope), we can use two points. Let's first consider the points at 0 ft and 50 ft:

\[
m = \frac{T(50) - T(0)}{50 - 0} = \frac{53 - 74}{50 - 0} = \frac{-21}{50} = -0.42
\]

Now that we have \( m = -0.42 \), we can use the point at \( d = 0 \) and \( T(0) = 74 \) to find \( b \):

\[
74 = -0.42 \cdot 0 + b \quad \Rightarrow \quad b = 74
\]

Thus, the equation is:

\[
T(d) = -0.42d + 74
\]

### Equation Check:

Let’s test this equation with a few values:
- At \( d = 0 \): \( T(0) = -0.42(0) + 74 = 74 \) (correct)
- At \( d = 50 \): \( T(50) = -0.42(50) + 74 = 53 \) (correct)

This linear equation fits the endpoints well, though there might be some deviation in the middle (especially between 20 and 30 ft). For a more precise model, a piecewise function or a different fitting approach (like quadratic) could be applied.

Would you like further refinement of the model?

### Follow-up Questions:
1. How does the temperature change per foot of depth between 20 ft and 30 ft?
2. Could a quadratic model better capture the sharp drop in temperature around 30 ft?
3. How would you modify this model if more data points were added?
4. Can this linear equation be used to predict temperature below 50 ft?
5. How could we improve this model if temperature changes became irregular at deeper levels?

### Tip:
When modeling real-world data, always check the residuals (differences between observed and predicted values) to see if a simple model like a linear one is sufficient, or if more complex models are needed.

Learn how to derive a linear equation that models the relationship between depth and temperature. The data includes depths of 0, 10, 20, 30, 40, and 50 feet, with corresponding temperatures of 74°F, 72°F, 71°F, 61°F, 55°F, and 53°F. This step-by-step solution will cover key mathematical concepts, such as the slope-intercept form, and is suitable for high school students.

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