Certainly! Let's analyze the given data points and predict the water levels in the Ogallala Aquifer for the next decade. Here are the provided data points for the years and the corresponding saturated thickness in feet:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Year} & \text{Saturated Thickness (feet)} \\
\hline
1975 & 107.5 \\
1980 & 95.5 \\
1985 & 84.25 \\
1990 & 73.75 \\
1995 & 63.75 \\
2000 & 55.25 \\
2005 & 47.75 \\
2010 & 40.25 \\
\hline
\end{array}
\][/tex]
We'll follow these steps to make our prediction:
1. Determine the total duration over which the data is provided:
The data spans from 1975 to 2010, so the total duration is:
[tex]\[
2010 - 1975 = 35 \text{ years}
\][/tex]
2. Calculate the total decrease in saturated thickness over this period:
The initial thickness in 1975 was 107.5 feet, and the final thickness in 2010 was 40.25 feet. The total decrease in thickness over the 35 years is:
[tex]\[
107.5 - 40.25 = 67.25 \text{ feet}
\][/tex]
3. Compute the average rate of decrease per year:
The average yearly decrease in saturated thickness is:
[tex]\[
\frac{67.25 \text{ feet}}{35 \text{ years}} \approx 1.9214 \text{ feet per year}
\][/tex]
4. Predict the water level for the year 2020:
To make a prediction for the year 2020, we need to determine how many years have passed since the last provided data point in 2010, which is:
[tex]\[
2020 - 2010 = 10 \text{ years}
\][/tex]
Using the average yearly decrease rate, we can find the predicted decrease in thickness over these 10 years:
[tex]\[
1.9214 \text{ feet per year} \times 10 \text{ years} = 19.214 \text{ feet}
\][/tex]
Therefore, the predicted saturated thickness in 2020 will be:
[tex]\[
40.25 \text{ feet} - 19.214 \text{ feet} = 21.036 \text{ feet}
\][/tex]
Thus, based on the data and our calculations, we predict that the saturated thickness of the Ogallala Aquifer will decrease to approximately [tex]\( 21.036 \)[/tex] feet by the year 2020. This prediction assumes that the rate of decrease remains constant over the next decade.
