Answer:
To find a polynomial function of degree 3 that models the given data, you can use polynomial regression. Polynomial regression fits a polynomial equation to the data that best represents the relationship between the independent variable (years, X) and the dependent variable (consumption, Y).
A) Polynomial Function of Degree 3:
The general form of a polynomial of degree 3 is:
�
=
�
�
3
+
�
�
2
+
�
�
+
�
Y=aX
3
+bX
2
+cX+d
To find the coefficients (a, b, c, and d), you can use a regression analysis tool or software. However, I'll walk you through the steps manually:
Create a system of equations using the data points:
For the data points (X, Y):
(1960, 12.4)
(1970, 21.8)
(1980, 20.4)
(1990, 19.3)
(2000, 24.0)
You'll have the following equations:
�
(
196
0
3
)
+
�
(
196
0
2
)
+
�
(
1960
)
+
�
=
12.4
a(1960
3
)+b(1960
2
)+c(1960)+d=12.4
�
(
197
0
3
)
+
�
(
197
0
2
)
+
�
(
1970
)
+
�
=
21.8
a(1970
3
)+b(1970
2
)+c(1970)+d=21.8
�
(
198
0
3
)
+
�
(
198
0
2
)
+
�
(
1980
)
+
�
=
20.4
a(1980
3
)+b(1980
2
)+c(1980)+d=20.4
�
(
199
0
3
)
+
�
(
199
0
2
)
+
�
(
1990
)
+
�
=
19.3
a(1990
3
)+b(1990
2
)+c(1990)+d=19.3
�
(
200
0
3
)
+
�
(
200
0
2
)
+
�
(
2000
)
+
�
=
24.0
a(2000
3
)+b(2000
2
)+c(2000)+d=24.0
Solve this system of equations to find the coefficients (a, b, c, and d). You can use a matrix solver or software like Python, R, or Excel to do this.
Once you have the coefficients, you'll have your polynomial function of degree 3.
B) Estimate Natural Gas Consumption in 1974 and 2010:
To estimate consumption in 1974 and 2010, simply plug these years (X values) into the polynomial function you found in part A:
For 1974 (X = 1974):
�
(
1974
)
=
�
(
197
4
3
)
+
�
(
197
4
2
)
+
�
(
1974
)
+
�
Y(1974)=a(1974
3
)+b(1974
2
)+c(1974)+d
For 2010 (X = 2010):
�
(
2010
)
=
�
(
201
0
3
)
+
�
(
201
0
2
)
+
�
(
2010
)
+
�
Y(2010)=a(2010
3
)+b(2010
2
)+c(2010)+d
Plug the values of a, b, c, and d that you found in part A into these equations to estimate the consumption in 1974 and 2010.
Explanation: