Question

The table shows the annual consumption of cheese per person in the United States for selected years in the twentieth century. Use a cubic model to estimate milk production in 1978.

Year | Pounds Consumed
1908 | 3.255
1937 | 9.053
1959 | 17.837
1996 | 58.395

Answer 1

The general form of a cubic model is given by:


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

Let x represent the number of years after 1900, then from the table, we have:


`3.225=(8)^3a+(8)^2b+8c+d \\ \\ \Rightarrow3.225=512a+64b+8c+d\ .\ .\ .\ (1) \\ \\ 9.035=(37)^3a+(37)^2b+37c+d \\ \\ \Rightarrow9.035=50653a+1369b+37c+d\ .\ .\ .\ (2) \\ \\ 17.837=(59)^3a+(59)^2b+59c+d \\ \\ 17.837=205379a+3481b+59c+d\ .\ .\ .\ (3) \\ \\ 58.395=(96)^3a+(96)^2b+96c+d \\ \\ \Rightarrow58.395=884736a+9216b+96c+d\ .\ .\ .\ (4)`

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

a = 0.00008956; b = -0.005398; c = 0.2884; d = 1.2175

Therefore, the cubic cubic model that estimates the annual consumption of cheese per person in the United States x years after 1900 is given by:


`f(x)=0.00008956x^3-0.005398x^2+0.2884x+1.2175`

Therefore, an estimate of the annual consumption of cheese per person in the United States in 1978 (78 years after 1900) is given by:


`0.00008956(78)^3-0.005398(78)^2+0.2884(78)+1.2175 \\ \\ =0.00008956(474552)-0.005398(6084)+22.4952+1.2175 \\ \\ =42.5009-32.8414+23.7127=33.3722`