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An underground water source: An underground aquifer near Seiling, Oklahoma, sits on an impermeable layer of limestone. West of Seiling the limestone layer is thought to slope downward in a straight line. In order to map the limestone layer, hydrologists started at Seiling heading west and drilled sample wells. Two miles west of Seiling the limestone layer was found at a depth of 220 feet. Three miles west of Seiling the limestone layer was found at a depth of 270 feet. a. What would you expect to be the depth of the limestone layer 5 miles west of Seiling? b. You want to drill a well down to the limestone layer as far west from Seiling as you can. Your budget will allow you to drill a well that is 290 feet deep. How far west of Seiling can you go to drill the well? c. Four miles west of Seiling someone drilled a well and found the limestone layer at 273 feet. Were the hydrologists right in saying that the limestone layer slopes downward in a straight line west of Seiling? Explain your reasoning.

Step by step solution

01

Understand the Problem

We need to find the equation of a line to model the depth of the limestone layer based on given data. We have data points: (2 miles, 220 feet) and (3 miles, 270 feet). From here, we can calculate the slope and form the equation to predict depths at other distances.

02

Calculate the Slope of the Line

The slope of the line is determined by the change in depth divided by the change in distance. Using the points \( (2, 220) \) and \( (3, 270) \), the slope is calculated as follows: \[ m = \frac{270 - 220}{3 - 2} = 50 \text{ feet per mile} \]

03

Determine the Linear Equation

Using the point-slope form of a line \( y - y_1 = m(x - x_1) \), and using the point \( (2, 220) \), the equation becomes: \[ y - 220 = 50(x - 2) \]Simplifying it, the equation of the line is \( y = 50x + 120 \).

04

Predict Depth at 5 Miles

Substitute \( x = 5 \) into the equation to find the depth at 5 miles. \[ y = 50(5) + 120 = 250 + 120 = 370 \text{ feet} \] Thus, at 5 miles west, the limestone layer would be expected at 370 feet.

05

Determine the Farthest Distance to Drill with Current Budget

Substitute \( y = 290 \) into the equation to find the maximum distance, \( x \), such that the depth doesn't exceed 290 feet. \[ 290 = 50x + 120 \]Solving for \( x \): \[ 50x = 290 - 120 = 170 \]\[ x = \frac{170}{50} = 3.4 \text{ miles} \] Thus, the farthest you can go is 3.4 miles west of Seiling.

06

Evaluate the Linear Assumption Conclusion

The limestone depth found 4 miles west is 273 feet. Using our linear equation for 4 miles, \[ y = 50(4) + 120 = 320 \text{ feet} \]. Since 273 feet is not equal to 320 feet, it shows the depth does not exactly match our linear model. This suggests the assumption of a perfectly linear slope may not be correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Calculation

Understanding how to calculate the slope is crucial in linear regression. The slope indicates how steep a line is and is calculated from two known data points. With the data from the problem: at 2 miles, the limestone is 220 feet deep, and at 3 miles, it's 270 feet deep. The difference in depth (rise) is 270 - 220, which equals 50 feet. The difference in distance (run) is 3 - 2, which equals 1 mile.
Therefore, the slope \( m \) is:

• \( m = \frac{50 \text{ feet}}{1 \text{ mile}} = 50 \text{ feet per mile} \)

This means for every mile you go west from Seiling, the limestone depth increases by 50 feet. Slope calculation is fundamental in predicting values in linear models, helping us understand relationships between variables.

Equation of a Line

Once the slope is determined, the next step is forming the equation of a line. The equation models how the limestone depth changes with distance from Seiling. We use the point-slope form, which is:

• \( y - y_1 = m(x - x_1) \)

Substitute \( m = 50 \), and one of the data points, say \( (2, 220) \), into the equation:

• \( y - 220 = 50(x - 2) \)

This simplifies to the equation:

• \( y = 50x + 120 \)

This is the equation of a line in the slope-intercept form \( y = mx + c \), where 50 represents the slope and 120 is the y-intercept. This linear equation lets us predict the limestone depth at any given distance west of Seiling.

Modeling Geological Data

Modeling real-world data often involves making assumptions that simplify the situation, like assuming linear behavior. With the geological data at hand, the goal is to predict the limestone depth at various distances using the linear model.We used the linear equation \( y = 50x + 120 \) to estimate depths at different points:

• At 5 miles, \( y = 370 \) feet
• To check budget constraints: if the max depth is 290 feet, solve \( 290 = 50x + 120 \), which gives \( x = 3.4 \) miles

However, real-world data is not always perfectly linear. For instance, the actual depth at 4 miles (273 feet) did not match our model's prediction (320 feet). This discrepancy suggests the model's assumption may not be entirely accurate, signalling the need for further investigation or a more complex model to account for variability in geological features. Understanding this helps refine models and improve predictions for better accuracy.