Problem 41 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The alternative minimum tax was created in 1969 to prevent the very wealthy from using creative deductions and shelters to avoid having to pay anything to the Internal Revenue Service. But it has increasingly hit the middle class. The number of taxpayers subjected to an alternative minimum tax is projected to be $$ N(t)=\frac{35.5}{1+6.89 e^{-0.8674 t}} \quad(0 \leq t \leq 6) $$ where \(N(t)\) is measured in millions and \(t\) is measured in years, with \(t=0\) corresponding to 2004 . What is the projected number of taxpayers subjected to an alternative minimum tax in 2010 ?

Step-by-Step Solution

Answer

The projected number of taxpayers subjected to an alternative minimum tax in 2010 is approximately \(34.19\) million.

Step 1: Write down the given formula for N(t)

The formula for the number of taxpayers subjected to an alternative minimum tax is: \[ N(t)=\frac{35.5}{1+6.89 e^{-0.8674 t}} \quad(0 \leq t \leq 6) \]

Step 2: Find the value of N(t) for t = 6

To find the projected number of taxpayers subjected to an alternative minimum tax in 2010, we need to evaluate N(t) at t = 6: \[ N(6)=\frac{35.5}{1+6.89 e^{-0.8674 (6)}} \]

Step 3: Evaluate the exponent and the term in the denominator

Calculate the exponent within the expression for N(6): \[ e^{-0.8674 (6)} = e^{-5.2044} \] Next, calculate the term in the denominator: \[ 1 + 6.89 e^{-5.2044} \approx 1 + 6.89 (0.0055) \approx 1.03795 \]

Step 4: Evaluate N(6)

Now, divide the numerator by the calculated denominator to find N(6): \[ N(6) = \frac{35.5}{1.03795} \approx 34.19 \]

Step 5: Interpret the result

The projected number of taxpayers subjected to an alternative minimum tax in 2010 is approximately 34.19 million.

Key Concepts

Applied Mathematics

When we utilize mathematics to solve real-world problems, such as examining the impact of tax policies on populations, we are engaging in applied mathematics. By utilizing mathematical equations and models, we can predict outcomes, identify trends, and make decisions based on logical, numerical analysis.

For instance, the alternative minimum tax (AMT) problem showcases how applied mathematics helps in understanding fiscal policies. Here, a mathematical formula—specifically, a rational function—is used to project the number of taxpayers affected by the AMT over time. This type of modeling allows for both quantitative analysis and the understanding of how legislative changes could influence socio-economic parameters.

Exponential Decay Model

The exponential decay model is commonly used in various fields like physics, finance, and epidemiology to describe how quantities decrease over time. The essential characteristic of this model is that the rate of decay is proportional to the quantity remaining, leading to a rapid initial decrease that slows over time.

In the exercise, the formula for the number of taxpayers affected by the AMT incorporates an exponential decay function, expressed as \( e^{-0.8674 t} \). This reflects how the increase in the number of taxpayers affected slows down as time passes, modeling a situation where initial growth is high but stabilizes as the effect of the tax regulation matures. It's essential to understand the nature of this decay to appreciate how it affects long-term projections for taxpayers.

Mathematical Modeling

Mathematical modeling involves creating mathematical representations of real-world systems to study their behavior. These models can be used for prediction, optimization, and understanding complex phenomena.

In the context of the alternative minimum tax problem, the mathematical model \( N(t) = \frac{35.5}{1+6.89 e^{-0.8674 t}} \) represents the number of taxpayers over time. The model captures the essence of the problem, which is to project the AMT's effect over six years. To get a reliable forecast, inputting the variable \( t \) corresponding to a specific year is crucial. Understanding and constructing such models is a vital skill in applied mathematics, enabling us to interpret and predict behaviors within defined systems.