Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The change from religious to lay teachers at Roman Catholic schools has been partly attributed to the decline in the number of women and men entering religious orders. The percentage of teachers who are lay teachers is given by $$ f(t)=\frac{98}{1+2.77 e^{-t}} \quad(0 \leq t \leq 4) $$ where \(t\) is measured in decades, with \(t=0\) corresponding to the beginning of 1960 . What percentage of teachers were lay teachers at the beginning of 1990 ?

Step-by-Step Solution

Answer

The percentage of lay teachers at Roman Catholic schools at the beginning of 1990 was approximately 86.1%.

Step 1: Substitute the value of t

We are given the function \(f(t) = \frac{98}{1 + 2.77e^{-t}}\), where \(t\) represents the time measured in decades since 1960. We want to find the percentage of lay teachers at the beginning of 1990, so we will substitute \(t = 3\) into the given function: \(f(3) = \frac{98}{1 + 2.77e^{-3}}\)

Step 2: Evaluate the expression

Now that we have substituted the value of a time for \(t\), we will evaluate the expression: \(f(3) = \frac{98}{1 + 2.77e^{-3}} = \frac{98}{1 + 2.77 \cdot (e^{-3})}\) We first need to compute the term in the exponent, so we get: \(e^{-3} \approx 0.0498\) Now we can substitute this value back into the expression: \(f(3) = \frac{98}{1 + 2.77 \cdot 0.0498} = \frac{98}{1 + 0.138}\) Now, we will add the terms in the denominator: \(f(3) = \frac{98}{1.138}\) Finally, we will divide 98 by 1.138 to find the percentage of lay teachers at the beginning of 1990: \(f(3) \approx 86.1\% \)

Step 3: Interpret the result

At the beginning of 1990, about 86.1% of teachers at Roman Catholic schools were lay teachers.

Key Concepts

Mathematical Modeling

Mathematical modeling is a powerful tool that allows us to represent real-world situations with mathematical formulas and concepts. In this exercise, we deal with an exponential function to model the change in teacher demographics at Roman Catholic schools over time. The function given is \( f(t) = \frac{98}{1 + 2.77 e^{-t}} \), where \( t \) represents time in decades since 1960. This model helps in understanding how the percentage of lay teachers changes over time by predicting it through calculations.

• The variable \( t \) is crucial as it correlates with specific years. For instance, \( t = 3 \) corresponds to the year 1990.
• Exponential functions are common in modeling growth and decay processes, reflecting how rapidly changes can occur.

By substituting different values of \( t\), we can project changes in the percentage of lay teachers at different times. This practical application of mathematical modeling provides insights into trends and helps educators and policymakers make informed decisions.

Percentage Calculation

Calculating percentages is an essential skill, especially in understanding models like the one used here. The problem calls for the calculation of the percentage of lay teachers using the provided function. When calculating percentages, we are often looking for a simple comparison with a base value of 100.

• In this scenario, the outcome of the function \( f(t) \) is a percentage, represented as a fraction of 98, which compares the number of lay teachers to the total teaching staff.
• The exponential component \( e^{-t} \) ensures that the function reflects the decrease in religious teachers over decades. The base number of 98 demonstrates the upper limit of the percentage achievable as \( t \) increases.

After substituting \( t = 3 \) in the exercise, the percentage was calculated by simplifying the equation: \( f(3) = \frac{98}{1 + 2.77 \cdot e^{-3}} \). This simplification and calculation led to the result of approximately 86.1% lay teachers by the beginning of 1990.

Applied Mathematics

Applied mathematics involves utilizing mathematical techniques to solve real-world problems, like predicting shifts in workforce composition using historical data. In this exercise, the exponential function is a form of applied mathematics used to forecast and analyze social changes within education systems.

• It bridges the gap between theoretical results and practical applications, offering models that help anticipate future developments.
• By interpreting the output of the function in context, educators can understand the dynamics affecting staffing in schools.

The accurate prediction of 86.1% of lay teachers in 1990 illustrates how applied mathematics makes seemingly complex social phenomena comprehensible and measurable. Furthermore, it emphasizes the importance of mathematical concepts to analyze trends and drive data-informed strategies.