To determine the percentage of women in the labor force for the year 2015 using the given model function:
[tex]\[ f(x) = \frac{66.23}{1 + 1.084 e^{-x / 24.77}} \][/tex]
we need to follow these steps:
### Step 1: Calculate [tex]\( x \)[/tex] for the year 2015
The variable [tex]\( x \)[/tex] represents the number of years since 1950. Therefore, for the year 2015:
[tex]\[ x = 2015 - 1950 \][/tex]
[tex]\[ x = 65 \][/tex]
### Step 2: Substitute [tex]\( x \)[/tex] into the function
Now, substitute [tex]\( x = 65 \)[/tex] into the given function [tex]\( f(x) \)[/tex]:
[tex]\[ f(65) = \frac{66.23}{1 + 1.084 e^{-65 / 24.77}} \][/tex]
### Step 3: Compute the value inside the function
First, calculate [tex]\(-\frac{65}{24.77}\)[/tex]:
[tex]\[ -\frac{65}{24.77} \approx -2.624 \][/tex]
Next, calculate the exponential part:
[tex]\[ e^{-2.624} \approx 0.0728 \][/tex]
Now substitute this value back into the denominator:
[tex]\[ 1 + 1.084 \cdot 0.0728 \approx 1 + 0.0789 \approx 1.0789 \][/tex]
### Step 4: Complete the function calculation
Finally, compute the entire function [tex]\( f(65) \)[/tex]:
[tex]\[ f(65) = \frac{66.23}{1.0789} \approx 61.404 \][/tex]
### Step 5: Round to the nearest whole number
To answer the question, round the calculated percentage to the nearest whole number:
[tex]\[
\boxed{61\%}
\][/tex]
Therefore, in 2015, approximately 61% of the labor force was comprised of women.
