The percent of women in the United States who were in the...

The percent of women in the United States who were in the civilian labor force increased rapidly for several decades and then stabilized. If $x$ represents the number of years since $1950,$ the function $$f(x)=\frac{67.21}{1+1.081 e^{-x / 24.71}}$$ models the percent fairly well. (Source: Monthly Labor Review, U.S. Bureau of Labor Statistics.) (a) What percent of U.S. women were in the civilian labor force in $2008 ?$ (b) In what year were $55 \%$ of U.S. women in the civilian labor force?

The percent of women in the United States who were in the civilian labor force increased rapidly for several decades and then stabilized. If $x$ represents the number of years since $1950,$ the function $$f(x)=\frac{67.21}{1+1.081 e^{-x / 24.71}}$$ models the percent fairly well. (Source: Monthly Labor Review, U.S. Bureau of Labor Statistics.)
(a) What percent of U.S. women were in the civilian labor force in $2008 ?$
(b) In what year were $55 \%$ of U.S. women in the civilian labor force?

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Key Concepts

Logistic Function

A logistic function is used to describe growth processes that start exponentially but then level off as they approach a saturation value. This type of function models the S-shaped, or sigmoidal, growth curve that is common in many real-world scenarios such as population growth, the spread of information, or market saturation.

Exponential Function

An exponential function describes processes where the rate of change is proportional to the current value, either growing or decaying rapidly. In the logistic model, an exponential term controls the early phase of rapid growth, influencing how quickly the function approaches its upper bound.

Parameter Substitution in Functions

Parameter substitution involves plugging specific values into a function to calculate the output. This technique is essential for determining the response of a modeled phenomenon at particular moments or under certain conditions, such as evaluating the percent value at a given time.

Solving Equations Using Logarithms

When a variable appears in the exponent, logarithms are used to isolate and solve for that variable. This method is key in inverse problems where you need to determine the time or other parameters corresponding to a specific output of a model, such as finding the year when a target percentage is reached.

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