To find a power model that expresses degrees Celsius ($T$) as a function of depth ($d$), we will assume the model has the form:

$T = a \cdot d^b$

where $T$ is the temperature in degrees Celsius, $d$ is the depth in kilometers, and $a$ and $b$ are constants to be determined.

Step 1: Transform the Data

To determine the constants $a$ and $b$, we can use logarithmic transformations. Taking the logarithm of both sides of the equation, we get:

$\ln(T) = \ln(a) + b \cdot \ln(d)$

This transformation allows us to perform a linear regression by letting:

• $y = \ln(T)$,
• $x = \ln(d)$,
• $\ln(a) = \text{intercept}$,
• $b = \text{slope}$.

Step 2: Calculate $\ln(T)$ and $\ln(d)$ for Each Data Pair

Using the data provided, we convert each $T$ and $d$ pair to its natural logarithm form.

Let's compute these values and then find the best-fit line for $\ln(T)$ as a function of $\ln(d)$ using linear regression. I will carry out the calculation now.The power model that best fits the data is:

$T = 47.679 \cdot d^{0.664}$

where:

• $T$ is the temperature in degrees Celsius,
• $d$ is the depth in kilometers.

This model represents the relationship between depth and temperature, with parameters rounded to three decimal places.

Would you like further details or have any questions?

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Here are some related questions you might consider:

1. How does a power model differ from a linear model in representing data?
2. Can this power model be used to predict temperatures at greater depths beyond the dataset?
3. What is the significance of the exponent $b$ in this power model?
4. How accurate is this model, and what are some potential sources of error?
5. How might the model change if new data points were added?

Tip: When creating models, always check the assumptions behind the model type to ensure it aligns well with the nature of your data.