Let's break down and solve the given problem step by step.
### Part (a)
First, we are to calculate the projected size of the civilian labor force for the year 2013. The given model to determine the labor force [tex]\( y \)[/tex] (in millions) for [tex]\( x \)[/tex] years after 1950 is:
[tex]\[ y = -0.0001x^3 + 0.0058x^2 + 1.43x + 56.7 \][/tex]
To find the labor force in 2013, we need to determine the value of [tex]\( x \)[/tex] in the context of the equation. Since [tex]\( x \)[/tex] represents the number of years after 1950:
[tex]\[ x = 2013 - 1950 = 63 \][/tex]
Substitute [tex]\( x = 63 \)[/tex] into the model:
[tex]\[ y = -0.0001(63)^3 + 0.0058(63)^2 + 1.43(63) + 56.7 \][/tex]
Now, let's evaluate this expression:
[tex]\[ y = -0.0001 \cdot 249057 + 0.0058 \cdot 3969 + 1.43 \cdot 63 + 56.7 \][/tex]
[tex]\[ y = -24.9057 + 23.4202 + 90.09 + 56.7 \][/tex]
[tex]\[ y = 144.8 \][/tex]
Therefore, the model projects the civilian labor force to be 144.8 million people in 2013.
### Part (b)
Next, we need to find the year when the model projects the civilian labor force to be 153 million people.
We have the equation:
[tex]\[ 153 = -0.0001x^3 + 0.0058x^2 + 1.43x + 56.7 \][/tex]
First, rearrange the equation for better clarity:
[tex]\[ -0.0001x^3 + 0.0058x^2 + 1.43x + 56.7 - 153 = 0 \][/tex]
[tex]\[ -0.0001x^3 + 0.0058x^2 + 1.43x - 96.3 = 0 \][/tex]
To find the value of [tex]\( x \)[/tex] that satisfies this equation, we can use numerical methods to solve for [tex]\( x \)[/tex].
Using the appropriate methods, it is determined that the value of [tex]\( x \)[/tex] which satisfies the equation is approximately [tex]\( x \approx 73 \)[/tex].
To find the corresponding year:
[tex]\[ \text{Year} = 1950 + x = 1950 + 73 = 2023 \][/tex]
Since the problem states to round up to the next year, the projected year when the labor force reaches 153 million is 2023.
### Summary
(a) The model projects the civilian labor force to be 144.8 million people in 2013.
(b) The model projects the civilian labor force to be 153 million people in the year 2023.
