Answer :
Sure, let's go through the data step-by-step and analyze it to predict the future changes in the thickness of the Ogallala Aquifer.
### Data Given
- Year: 1975, 1980, 1985, 1990, 1995, 2000, 2005, 2010
- Thickness (feet): 107.5, 95.5, 84.25, 73.75, 63.75, 55.25, 47.75, 40.25
### Step 1: Fit a Linear Regression Model
To understand the trend of the decline in the water levels, we use linear regression which fits a linear equation to the observational data. The general form of the equation is:
[tex]\[ \text{Thickness} = \text{slope} \times \text{Year} + \text{intercept} \][/tex]
From fitting the data to a linear model, we obtain:
- Slope: \( -1.9202380952380258 \)
- Intercept: \( 3897.0744047617636 \)
The slope here is negative, which indicates that the thickness of the aquifer's water levels is decreasing over time.
### Step 2: Predict Future Thickness
#### For the Year 2020
We plug the year 2020 into our regression equation to predict the saturated thickness:
[tex]\[ \text{Thickness}_{2020} = (-1.9202380952380258) \times 2020 + 3897.0744047617636 \][/tex]
Doing the math, we get:
[tex]\[ \text{Thickness}_{2020} = 18.193452380951385 \text{ feet} \][/tex]
#### For the Year 2030
Similarly, for the year 2030:
[tex]\[ \text{Thickness}_{2030} = (-1.9202380952380258) \times 2030 + 3897.0744047617636 \][/tex]
Doing the math, we get:
[tex]\[ \text{Thickness}_{2030} = -1.0089285714288962 \text{ feet} \][/tex]
### Interpretation
- Year 2020: The predicted saturated thickness is approximately 18.19 feet.
- Year 2030: The thickness becomes negative, implying that at this rate, the water level is expected to be depleted.
### Conclusion
Based on the analysis of the data and the predictions from the linear regression model, the water level in the Ogallala aquifer is expected to continue decreasing over the next decade. If the current trend continues, the aquifer could be nearly depleted by 2030. This indicates a serious depletion of the water resources, which needs to be addressed to ensure sustainable use of the aquifer.
### Data Given
- Year: 1975, 1980, 1985, 1990, 1995, 2000, 2005, 2010
- Thickness (feet): 107.5, 95.5, 84.25, 73.75, 63.75, 55.25, 47.75, 40.25
### Step 1: Fit a Linear Regression Model
To understand the trend of the decline in the water levels, we use linear regression which fits a linear equation to the observational data. The general form of the equation is:
[tex]\[ \text{Thickness} = \text{slope} \times \text{Year} + \text{intercept} \][/tex]
From fitting the data to a linear model, we obtain:
- Slope: \( -1.9202380952380258 \)
- Intercept: \( 3897.0744047617636 \)
The slope here is negative, which indicates that the thickness of the aquifer's water levels is decreasing over time.
### Step 2: Predict Future Thickness
#### For the Year 2020
We plug the year 2020 into our regression equation to predict the saturated thickness:
[tex]\[ \text{Thickness}_{2020} = (-1.9202380952380258) \times 2020 + 3897.0744047617636 \][/tex]
Doing the math, we get:
[tex]\[ \text{Thickness}_{2020} = 18.193452380951385 \text{ feet} \][/tex]
#### For the Year 2030
Similarly, for the year 2030:
[tex]\[ \text{Thickness}_{2030} = (-1.9202380952380258) \times 2030 + 3897.0744047617636 \][/tex]
Doing the math, we get:
[tex]\[ \text{Thickness}_{2030} = -1.0089285714288962 \text{ feet} \][/tex]
### Interpretation
- Year 2020: The predicted saturated thickness is approximately 18.19 feet.
- Year 2030: The thickness becomes negative, implying that at this rate, the water level is expected to be depleted.
### Conclusion
Based on the analysis of the data and the predictions from the linear regression model, the water level in the Ogallala aquifer is expected to continue decreasing over the next decade. If the current trend continues, the aquifer could be nearly depleted by 2030. This indicates a serious depletion of the water resources, which needs to be addressed to ensure sustainable use of the aquifer.
