Answer-
The cubic equation-
[tex]f(x)=0.00009121x^3-0.00666099x^2+0.427014x+1.36845[/tex]
Milk consumption in 1994 was 58.4 pounds.
Solution-
The general form of a cubic model is,
[tex]f(x)=ax^3+bx^2+cx+d[/tex]
x = input variable = year - 2000 i.e number years after 2000
y = output variable = milk consumed in pounds
Then putting the values from the table,
[tex]2.592=a(3)^3+b(3)^2+c(3)+d[/tex]
[tex]\Rightarrow 2.592=27a+9b+3c+d[/tex] --------------------------1
[tex]13.301=a(39)^3+b(39)^2+c(39)+d[/tex]
[tex]\Rightarrow 13.301=59319a+1521b+39c+d[/tex] -------------2
[tex]20.41=a(56)^3+b(56)^2+c(56)+d[/tex]
[tex]\Rightarrow 20.41=175616a+3136b+56c+d[/tex] -------------3
[tex]63.359=a(97)^3+b(97)^2+c(97)+d[/tex]
[tex]\Rightarrow 63.359=912673a+9409b+97c+d[/tex] ------------4
Solving equations (1), (2), (3) and (4) simultaneously we get,
[tex]a=0.00009121,\\\\b=-0.00666099,\\\\c=0.427014,\\\\d=1.36845[/tex]
So, the equation becomes,
[tex]f(x)=0.00009121x^3-0.00666099x^2+0.427014x+1.36845[/tex]
In order to get the milk consumption in 1994, putting x=94
[tex]=0.00009121(94)^3-0.00666099(94)^2+0.427014(94)+1.36845[/tex]
[tex]=(0.00009121)\cdot (830584)-(0.00666099)\cdot (8836)+(0.427014)\cdot (94)+(1.36845)[/tex]
[tex]=58.4088[/tex]
Therefore, milk consumption in 1994 was 58.4 pounds.
