To create a cubic model [tex]\( y \)[/tex] representing the number of women in the workforce based on years past January 1st, 1950, we will use the given data from the table. The model takes the form:
[tex]\[ y = Ax^3 + Bx^2 + Cx + D \][/tex]
where:
- [tex]\( x \)[/tex] is the number of years past 1950,
- [tex]\( y \)[/tex] is the number of women in the workforce (in millions),
- [tex]\( A, B, C, D \)[/tex] are coefficients to be determined.
Using the data provided, the years and corresponding workforce numbers can be listed as follows:
- 1950: [tex]\( x = 0, \, y = 18.9 \)[/tex]
- 1960: [tex]\( x = 10, \, y = 22.7 \)[/tex]
- 1970: [tex]\( x = 20, \, y = 31.5 \)[/tex]
- 1980: [tex]\( x = 30, \, y = 45.2 \)[/tex]
- 1990: [tex]\( x = 40, \, y = 55.6 \)[/tex]
- 2000: [tex]\( x = 50, \, y = 64.8 \)[/tex]
- 2010: [tex]\( x = 60, \, y = 75.7 \)[/tex]
- 2015: [tex]\( x = 65, \, y = 77.9 \)[/tex]
- 2020: [tex]\( x = 70, \, y = 78.9 \)[/tex]
- 2030: [tex]\( x = 80, \, y = 80.8 \)[/tex]
- 2040: [tex]\( x = 90, \, y = 85.1 \)[/tex]
- 2050: [tex]\( x = 100, \, y = 92.3 \)[/tex]
Using these data points, we fit a cubic polynomial to determine the coefficients [tex]\( A, B, C, \)[/tex] and [tex]\( D \)[/tex]. The fitted cubic model is:
[tex]\[ y = -7.0 \times 10^{-5} x^3 + 0.00565 x^2 + 0.83946 x + 16.23384 \][/tex]
Thus, the cubic model [tex]\( y \)[/tex] can be written as:
[tex]\[ y = -0.00007 x^3 + 0.00565 x^2 + 0.83946 x + 16.23384 \][/tex]
So, in the correct format:
[tex]\[ y = -0.00007 x^3 + 0.00565 x^2 + 0.83946 x + 16.23384 \][/tex]
