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The following table gives the number of women aged 16 years and older (in millions) in a country's civilian workforce for selected years from 1950 and projected to 2050.

Complete parts (a) and (b) below.

\begin{tabular}{|c|c|c|c|}
\hline Year & \begin{tabular}{c}
Women in the \\
Workforce (millions)
\end{tabular} & Year & \begin{tabular}{c}
Women in the \\
Workforce (millions)
\end{tabular} \\
\hline 1950 & 18.9 & 2010 & 75.7 \\
\hline 1960 & 22.7 & 2015 & 77.9 \\
\hline 1970 & 31.5 & 2020 & 78.9 \\
\hline 1980 & 45.2 & 2030 & 80.8 \\
\hline 1990 & 55.6 & 2040 & 85.1 \\
\hline 2000 & 64.8 & 2050 & 92.3 \\
\hline
\end{tabular}

a. Use [tex]$x$[/tex] as the number of years past January 1st, 1950 to create a cubic model, [tex]$y$[/tex], using these data.

[tex]\[ y = (\square)x^3 + (\square)x^2 + (\square)x + (\square) \][/tex]

(Type integers or decimals rounded to five decimal places.)

Sagot :

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]
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