<PRE><B>Question 113530</B>
Fitting a quadratic model to data
Throughout the twentieth century, the eating habits of Americans changed noticeably .Americans started consuming less whole milk and butter and started consuming more skim and low fat milk and margarine. We also started eating more poultry and fish. In this project you will have the opportunity to investigate trends in per capita consumption of poultry during the twentieth century .
We will start by finding a quadratic model  y=ax^2+bx+c  that has ordered pair solutions that correspond to the data for US. Per capita consumption of poultry given in the table .To do so, substitute each data pair into the equation. Each time, the result is an equation in three unknown: a,b, and c. Because there are three pair of data, we can form a system of three linear equation equations in three unknowns .Solving for the values of a, b, and c gives a quadratic model that represents the given data.
US PER CAPITA CONSUMPTION OF POULTRY (IN POUNDS)
YEAR                         X                            POULTRY CONSUMPTION Y (IN POUNDS)
1909                         9                                11
1969                        69                              33
2001                        101                            66
	Write the system of equations that must be solved to find the values of a, b, and c needed for a quadratic model of the given data.
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y = ax^2+bx+c 
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11 = 81a +9b +c
33 = 69^2a + 69b +c
66 = 101^2a + 101b +c
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	Solve the system of equations for a, b c.
a=0.0072237319...
b=-0.1967844203...
c=12.1859375...
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	Write the quadratic model for the date. Note that the variable x represents the number of years after 1990.
y = 0.0072237...x^2-0.1967844...x+12.185937....
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	In 1939 the actual U.S per capita consumption of poultry was 12 lbs per person. Based on this information, how accurate do you think this model is for years other than those given in the table?
1939-1990=-51
f(-51) = 41.011
The model tends to be more and more inaccurate for years further and 
further away from 1990.
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	 Use your model to estimate per capita consumption of poultry in 1950.
1950-1990 = -40
f(-40) = 31.615 obs
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	According to the model, in what year was per capita consumption of poultry 50 pounds per person?
f(-60) is approximately 50 lbs.
x=-60 corresponds to 1990-60 = 1930
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	In what year was the per capita consumption of poultry at its lowest level? 
f(17) = 10.928
x=17 corresponds to 1990+17 = 2007
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What was that level? 
f(17) = 10.928 lbs
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Cheers,
Stan H. 
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