Certainly! Let's work through the problem step-by-step to find the percent of women in a country's civilian labor force in the year 2012 using the given function [tex]\( f(x) = \frac{67.21}{1 + 1.085 e^{-x / 24.71}} \)[/tex].
### Step-by-Step Solution:
1. Identify the variable [tex]\( x \)[/tex]:
The variable [tex]\( x \)[/tex] represents the number of years since 1950. We need to find [tex]\( x \)[/tex] for the year 2012.
[tex]\[
x = 2012 - 1950 = 62
\][/tex]
2. Substitute [tex]\( x \)[/tex] into the function:
Now that we know [tex]\( x \)[/tex] is 62, we substitute it into the given function to calculate [tex]\( f(x) \)[/tex].
[tex]\[
f(62) = \frac{67.21}{1 + 1.085 e^{-62 / 24.71}}
\][/tex]
3. Calculate the expression within the function:
We need to compute the value of [tex]\( e^{-62 / 24.71} \)[/tex]. Let's denote this exponential calculation as [tex]\( e \)[/tex].
[tex]\[
e^{-62 / 24.71} \approx 0.03689 \quad (\text{approximate value for clarity})
\][/tex]
4. Calculate the denominator:
Add 1.085 times this exponential value to 1.
[tex]\[
1 + 1.085 \cdot 0.03689 \approx 1.04004
\][/tex]
5. Calculate the function value:
Now compute the value of [tex]\( f(x) \)[/tex].
[tex]\[
f(62) = \frac{67.21}{1.04004} \approx 61.76
\][/tex]
6. Round to the nearest whole number:
Finally, we round the computed value to the nearest whole number.
[tex]\[
61.76 \approx 62
\][/tex]
Therefore, the percent of women in the labor force in 2012, rounded to the nearest whole number, is:
[tex]\[
\boxed{62\%}
\][/tex]
