Question

From 2000 through 2005, the average salary for public school nurses \(S\) (in dollars) in the United States changed at the rate of \(\frac{d S}{d t}=1724.1 e^{-t / 4.2}\) where \(t=0\) corresponds to \(2000 .\) In 2005, the average salary for public school nurses was \(\$ 40,520 .\) (Source: Educational Research Service) (a) Write a model that gives the average salary for public school nurses per year. (b) Use the model to find the average salary for public school nurses in 2002 .

Short answer

The model for the average salary for public school nurses per year is \( S(t) = -8157e^{-t/4.2} + 49013 \). The average salary in 2002 was approximately \$44938.

Step-by-step solution

Integral Calculation

The first step is to calculate the integral of the function \( dS/dt = 1724.1e^{-t/4.2} \) to find the equation for \( S \). The result is \( S(t) = -8157e^{-t/4.2} + C \), where \( C \) is the constant of integration.

Solving for the Constant of Integration

Next, substitute values for \( S \) and \( t \) from the year 2005 to find the constant of integration \( C \). That is, when \( t = 5 \) (2005 is 5 years from 2000), \( S = 40520 \). This gives \( 40520 = -8157e^{-5/4.2} + C \). Solving this for \( C \) gives \( C \approx 49013 \). Thus, the model for the average salary is \( S(t) = -8157e^{-t/4.2} + 49013 \)

Average Salary in 2002

Now, to find the average salary in 2002, substitute \( t = 2 \) into the model. This gives \( S(2) = -8157e^{-2/4.2} + 49013 \), which approximates to \$44938.

Key concepts

Average Salary

The concept of average salary, particularly in the context of public school nurses from 2000 to 2005, is an example of how changes over time can be analyzed using calculus. To understand how average salaries vary year by year, we look at the rate at which they change over time. This rate is given by the derivative function, which in this problem, is expressed as \( \frac{dS}{dt} = 1724.1 e^{-t/4.2} \). By using this derivative, we can determine how the salary increases over time, starting from a known point.
In this exercise, the exact average salary for 2005 is given as \$40,520. This benchmark allows us to use calculus to find an equation that represents the average salary for any year in the interval. This makes it clear that the past data and current rates of change (from the derivative) provide insights into future trends.
Students also learn that economics and daily problems like salary analysis can be tackled using mathematics, showcasing the practical applications of calculus for modeling financial and economic patterns.

Integration

Integration is a fundamental concept in calculus that helps us find an original function when we are provided with its rate of change, which is known as its derivative. In this instance, the average salary rate of change is given by the function \( \frac{dS}{dt} = 1724.1 e^{-t/4.2} \), which we need to integrate to find the function \( S(t) \) describing the salary itself.
The integration process involves finding the antiderivative. For this problem, integrating \( 1724.1 e^{-t/4.2} \) gives us the function \( S(t) = -8157e^{-t/4.2} + C \), where \( C \) is the constant of integration.
This constant is vital because it reflects initial conditions or specific known values, like the salary at a particular year, rather than just changes in value. It is typically found by substituting known values into the integrated function, like using the known salary in 2005 to solve for \( C \). Thus, integration not only helps in recreating the salary function but also firmly anchors functions in real-world data points.

Modeling Functions

Modeling functions is a calculus concept that entails creating mathematical representations of real-world scenarios. In this exercise, the model we derive is about the average salary of public school nurses over a period. Functions are used to predict unknown values based on known variables and rates of change.
By starting with the derivative, \( \frac{dS}{dt} \), which reflects the rate of salary change, we integrate to form a model, \( S(t) = -8157e^{-t/4.2} + 49013 \). This model allows us to calculate the average salary at any given year without needing precise data for each year.

• The function takes into account real changes, like how economic factors might affect salary growth.
• It simplifies complexity by using equations to express potentially erratic trends more smoothly.

Students learning to model functions study how various assumptions and rates tie into final outcomes, highlighting how predictive tools in calculus help make sense of uncertain futures.