To predict the saturated thickness of water in the year 2015, we'll analyze the historical data given in the table and determine the average rate of decrease in thickness over the years. Here are the steps broken down:
1. Identify the given data:
- Years: 1975, 1980, 1985, 1990, 1995, 2000, 2005, 2010.
- Thickness in meters: 32.77, 29.11, 25.68, 22.48, 19.43, 16.84, 14.55, 12.27.
2. Calculate the total decrease in thickness from 1975 to 2010:
[tex]\[
\text{Total decrease} = \text{Initial thickness (in 1975)} - \text{Final thickness (in 2010)}
\][/tex]
[tex]\[
\text{Total decrease} = 32.77\, m - 12.27\, m = 20.50\, m
\][/tex]
3. Find the time span from 1975 to 2010:
[tex]\[
\text{Total years} = 2010 - 1975 = 35\, years
\][/tex]
4. Determine the average rate of decrease in thickness per year:
[tex]\[
\text{Average decrease per year} = \frac{\text{Total decrease}}{\text{Total years}}
\][/tex]
[tex]\[
\text{Average decrease per year} = \frac{20.50\, m}{35\, years} \approx 0.5857\, m/year
\][/tex]
5. Predict the thickness for the year 2015:
- Number of years from 2010 to 2015:
[tex]\[
\text{Years passed} = 2015 - 2010 = 5\, years
\][/tex]
- Decrease in thickness from 2010 to 2015:
[tex]\[
\text{Decrease} = \text{Years passed} \times \text{Average decrease per year}
\][/tex]
[tex]\[
\text{Decrease} = 5\, years \times 0.5857\, m/year = 2.9286\, m
\][/tex]
- Predicted thickness in 2015:
[tex]\[
\text{Predicted thickness} = \text{Thickness in 2010} - \text{Decrease}
\][/tex]
[tex]\[
\text{Predicted thickness} = 12.27\, m - 2.9286\, m \approx 9.34\, m
\][/tex]
Based on these calculations, the saturated thickness in 2015 is predicted to be approximately 9.34 meters.
Therefore, the correct prediction for the saturated thickness in 2015 is:
- It will be near 10 meters.
