Answer :
To determine the saturated thickness for the year 2020, we can use the information given in the question, particularly the data on the saturated thickness of the aquifer over a range of years. This is a straightforward application of linear regression, which helps us identify a trend and predict future values based on that trend.
Here are the steps involved:
1. Identify the Data Points:
- We have the years: \(1975, 1980, 1985, 1990, 1995, 2000, 2005, 2010\).
- Corresponding saturated thickness values: \(32.77, 29.11, 25.68, 22.48, 19.43, 16.84, 14.55, 12.27\).
2. Establish the Linear Relationship:
- Using linear regression, we determine the best-fit line through these data points. This line can be represented with the equation:
[tex]\[ thickness = slope \times year + intercept \][/tex]
- The slope (\(m\)) and intercept (\(b\)) for our line are derived from the data.
3. Calculate the Slope and Intercept:
- Based on the provided solution, we have:
[tex]\[ \text{slope} = -0.585 \, (\text{approximately}), \][/tex]
[tex]\[ \text{intercept} = 1188.06 \, (\text{approximately}). \][/tex]
4. Predicting the Saturated Thickness in 2020:
- Using the linear equation derived:
[tex]\[ \text{saturated thickness in 2020} = -0.585 \times 2020 + 1188.06 \][/tex]
- Performing the calculation:
[tex]\[ \text{saturated thickness in 2020} \approx -0.585 \times 2020 + 1188.06 \approx 5.54 \, \text{meters} \][/tex]
5. Conclusion:
- Based on our calculations, the saturated thickness in 2020 will be approximately \(5.54\) meters.
- Since [tex]\(5.54\)[/tex] meters falls below the range of the lowest given option (8-10 meters), none of the given options accurately represent this value. However, lacking any corrections, we can confidently state the precise value calculated is around [tex]\(5.54\)[/tex] meters.
Here are the steps involved:
1. Identify the Data Points:
- We have the years: \(1975, 1980, 1985, 1990, 1995, 2000, 2005, 2010\).
- Corresponding saturated thickness values: \(32.77, 29.11, 25.68, 22.48, 19.43, 16.84, 14.55, 12.27\).
2. Establish the Linear Relationship:
- Using linear regression, we determine the best-fit line through these data points. This line can be represented with the equation:
[tex]\[ thickness = slope \times year + intercept \][/tex]
- The slope (\(m\)) and intercept (\(b\)) for our line are derived from the data.
3. Calculate the Slope and Intercept:
- Based on the provided solution, we have:
[tex]\[ \text{slope} = -0.585 \, (\text{approximately}), \][/tex]
[tex]\[ \text{intercept} = 1188.06 \, (\text{approximately}). \][/tex]
4. Predicting the Saturated Thickness in 2020:
- Using the linear equation derived:
[tex]\[ \text{saturated thickness in 2020} = -0.585 \times 2020 + 1188.06 \][/tex]
- Performing the calculation:
[tex]\[ \text{saturated thickness in 2020} \approx -0.585 \times 2020 + 1188.06 \approx 5.54 \, \text{meters} \][/tex]
5. Conclusion:
- Based on our calculations, the saturated thickness in 2020 will be approximately \(5.54\) meters.
- Since [tex]\(5.54\)[/tex] meters falls below the range of the lowest given option (8-10 meters), none of the given options accurately represent this value. However, lacking any corrections, we can confidently state the precise value calculated is around [tex]\(5.54\)[/tex] meters.
