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The percent of women in a country's civilian labor force can be modeled fairly well by the function [tex]f(x)=\frac{66.23}{1+1.084 e^{-x / 24.77}}[/tex], where [tex]x[/tex] represents the number of years since 1950.

Answer parts (a) and (b).

(a) In 2015, what percent, to the nearest whole number, of the labor force was comprised of women?

(Do not round until the final answer. Then round to the nearest integer as needed.)

Sagot :

GinnyAnsweridn
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.
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