The percentage of married women who earned more than their...

The percentage of married women who earned more than their husbands over the period from $1960(t=0)$ through $2011(t=51)$ was growing at the rate of approximately $$r(t)=-0.00025142 t^{2}+0.02116 t+0.0328 \quad(0 \leq t \leq 51)$$ percent/year in year $t$. The percentage of married women earning more than their husbands in 1960 stood at 6.2\%. a. Find an expression giving the approximate percentage of married women who earned more than their husbands in year $t$ b. Use the result of part (a) to estimate the percentage of married women who earned more than their husbands in 2013 , assuming that the trend continued.

The percentage of married women who earned more than their husbands over the period from $1960(t=0)$ through $2011(t=51)$ was growing at the rate of approximately
$$r(t)=-0.00025142 t^{2}+0.02116 t+0.0328 \quad(0 \leq t \leq 51)$$
percent/year in year $t$. The percentage of married women earning more than their husbands in 1960 stood at 6.2\%.
a. Find an expression giving the approximate percentage of married women who earned more than their husbands in year $t$
b. Use the result of part (a) to estimate the percentage of married women who earned more than their husbands in 2013 , assuming that the trend continued.

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Key Concepts

Using Mathematical Models for Extrapolation

Once an expression for the percentage is obtained, it can be used to estimate values for years beyond the interval of the data, assuming the trend continues. This extrapolation is a common technique in applied mathematics for forecasting based on historical trends.

Initial Value and Its Role in Solving Differential Equations

The initial value, which is the percentage at the starting time, is crucial for determining the constant of integration. In differential equations, knowing the value of the function at a specific point allows for the particular solution to be identified.

Rate Function

A rate function expresses how a quantity changes with respect to time. In this context, it represents the yearly change in the percentage of married women earning more than their husbands, and integrating this function gives the cumulative change over a period.

Integration to Recover an Accumulated Value

Integrating the rate function allows us to find the accumulated percentage over time. This process involves finding the antiderivative of the rate function and then adding the initial percentage to determine the overall function that approximates the percentage at any given time.

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