Question

PLEASE HELP!!! URGENT

The table shows the annual consumption of cheese per person in the U.S. for selected years in the 20th century.

Use a cubic model to estimate milk production in 1994.

A. 62.4
B. 58.4
C. 116.8
D. 54.4

Answer 1

The cubic equation-

`f(x)=0.00009121x^3-0.00666099x^2+0.427014x+1.36845`

Milk consumption in 1994 was 58.4 pounds.

Solution-

The general form of a cubic model is,

`f(x)=ax^3+bx^2+cx+d`

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,

`2.592=a(3)^3+b(3)^2+c(3)+d`

`\Rightarrow 2.592=27a+9b+3c+d` --------------------------1

`13.301=a(39)^3+b(39)^2+c(39)+d`

`\Rightarrow 13.301=59319a+1521b+39c+d` -------------2

`20.41=a(56)^3+b(56)^2+c(56)+d`

`\Rightarrow 20.41=175616a+3136b+56c+d` -------------3

`63.359=a(97)^3+b(97)^2+c(97)+d`

`\Rightarrow 63.359=912673a+9409b+97c+d` ------------4

Solving equations (1), (2), (3) and (4) simultaneously we get,

`a=0.00009121,\\\\b=-0.00666099,\\\\c=0.427014,\\\\d=1.36845`

So, the equation becomes,

`f(x)=0.00009121x^3-0.00666099x^2+0.427014x+1.36845`

In order to get the milk consumption in 1994, putting x=94

`=0.00009121(94)^3-0.00666099(94)^2+0.427014(94)+1.36845`

`=(0.00009121)\cdot (830584)-(0.00666099)\cdot (8836)+(0.427014)\cdot (94)+(1.36845)`

`=58.4088`

Therefore, milk consumption in 1994 was 58.4 pounds.