Question

. During droughts, water for irrigation is pumped from the ground. When ground water is pumped excessively, the water table lowers. In California, lowering water tables have been linked to reduced water quality and sinkholes. A particular well in California’s Inland Empire has been monitored over many years. In 1994, the water level was 250 feet below the land surface. In 2000, the water level was 261 feet below the surface. In 2006, the water level was 268 feet below the surface. In 2009, the water level was 271 feet below the surface. In 2012, it was 274 feet below the surface. And in 2015, it was 276 feet below the surface. (a) Find a cubic (degree 3) polynomial model for this data on water level. First, define what x and y mean here, and write the data points you use. Then, find a cubic which is a best fit for this data, in the least squares sense. (b) Use your model to predict the water level of the well, in feet below the surface, in 2020. You may assume that the trends of 1994 to 2015 continue to 2020.

Answer 1

Year	Water Level (feet)
1994	250
2000	261
2006	268
2009	271
2012	274
2015	276

Here,

X = Year, Y= Water Level (feet)

Let 1990 be base so, x= 0

X	4	10	16	19	22	25
Y	250	261	268	271	274	276

Least squares method using matrix method:

Now, for the values provided for X and y, we get that the vector with estimated leat squares coefficients is computed as follows:

Therefore, based on the data provided, the estimated least square cubic (degree 3) polynomial equation is: (Rounded off to 4 decimals)

b)

Here 2020 year, X=30

Put X=30 in above equation,