A geologist has gathered the following data from a geophysical study and laboratory experiments to attempt to create a model of a particular are that she was assigned to study. She wants to create a model to determine the pressure-depth. Find a power model that express degrees Celsius as a function of the Depth.

Round your answers to 3 decimal places

Depth (km)	degrees Celsius
0.7	37.6
1	47.6
1.3	56.8
1.6	65.7
1.9	72.3
2.2	81
2.5	87.4


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.

A geologist has gathered the following data from a geophysical study and laboratory experiments to attempt to create a model of a particular area that she was assigned to study. She wants to create a model to determine the pressure-depth. Find a power model that expresses degrees Celsius as a function of the Depth. Round your answers to 3 decimal places.

Depth (km)    degrees Celsius
0.7    37.6
1    47.6
1.3    56.8
1.6    65.7
1.9    72.3
2.2    81
2.5    87.4

Discover how to model temperature as a function of depth using a power model based on a geophysical study. Given depth and temperature data, learn to apply logarithmic transformations and linear regression to find constants for the power model T = a * d^b, suitable for students in Grades 11-12.

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