Problem 70 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

AGING PoPULATION The population of Americans age \(55 \mathrm{yr}\) and older as a percentage of the total population is approximated by the function $$ f(t)=10.72(0.9 t+10)^{0.3} \quad(0 \leq t \leq 20) $$ where \(t\) is measured in years, with \(t=0\) corresponding to the year 2000 . At what rate was the percentage of Americans age 55 and over changing at the beginning of \(2000 ?\) At what rate will the percentage of Americans age \(55 \mathrm{yr}\) and older be changing in \(2010 ?\) What will be the percentage of the population of Americans age \(55 \mathrm{yr}\) and older in 2010 ?

Step-by-Step Solution

Answer

At the beginning of 2000, the percentage of Americans age 55 and older was increasing at a rate of about 1.064% per year. In 2010, the percentage will be increasing at a rate of about 0.403% per year. The percentage of the population of Americans age 55 and older in 2010 will be around 22.04%.

Step 1: Find the derivative of the function f(t)

To find the rate at which the percentage of Americans age 55 and older is changing, we need to find the first derivative of the function f(t), f'(t). We'll use the chain rule for this. Given function f(t) as: \(f(t)=10.72(0.9t+10)^{0.3}\) Differentiating this with respect to t, we have: \(f'(t) = 10.72 \cdot 0.3 (0.9t + 10)^{-0.7} \cdot (0.9)\) Simplifying, we get: \(f'(t) = 3.216 (0.9t + 10)^{-0.7} \)

Step 2: Evaluate the rate of change at the beginning of 2000 (t=0)

To find the rate at which the percentage of Americans age 55 and older was changing at the beginning of 2000, we need to evaluate the derivative f'(t) at t = 0: \(f'(0) = 3.216 (0.9 \cdot 0 + 10)^{-0.7}\) \(f'(0) = 3.216 (10)^{-0.7}\) Evaluating this, we get: \(f'(0) \approx 1.064\) So, at the beginning of 2000, the percentage of Americans age 55 and older was increasing at a rate of about 1.064% per year.

Step 3: Evaluate the rate of change in 2010 (t=10)

To find the rate at which the percentage of Americans age 55 and older will be changing in 2010, we need to evaluate the derivative f'(t) at t = 10: \(f'(10) = 3.216 (0.9 \cdot 10 + 10)^{-0.7}\) \(f'(10) = 3.216 (19)^{-0.7}\) Evaluating this, we get: \(f'(10) \approx 0.403\) So, in 2010, the percentage of Americans age 55 and older will be increasing at a rate of about 0.403% per year.

Step 4: Find the percentage of the population of Americans age 55 and older in 2010

To find the percentage of the population of Americans age 55 and older in 2010, we need to evaluate the given function f(t) at t=10: \(f(10) = 10.72(0.9 \cdot 10 + 10)^{0.3}\) \(f(10) = 10.72(19)^{0.3}\) Evaluating this, we get: \(f(10) \approx 22.04\) Hence, in 2010, the percentage of the population of Americans age 55 and older will be around 22.04%.

Key Concepts

Demographic Modeling

Demographic modeling is a technique utilized to represent and predict changes in population statistics over time. In our exercise related to the aging population, demographic modeling is critical in determining how the percentage of Americans aged 55 and older changes. Using the function provided, we can derive useful insights into population dynamics.
This statistical approach often involves creating mathematical functions that are informed by historical data, which in our case, is represented by \( f(t) = 10.72(0.9t+10)^{0.3} \). These models enable researchers, policymakers, and businesses to forecast future population trends and prepare accordingly. Here are some reasons why demographic modeling is essential:

• Helps in resource allocation and public service planning.
• Aids policy makers in understanding the need for changes in healthcare, social security, and other aging-related needs.
• Provides insights for economic planning and labor market adjustments.

Calculus Applications

Calculus is fundamental in uncovering trends and behaviors within demographic data, such as the rate change or predicting future scenarios. When dealing with the population function given in our task, calculus allows us to derive and understand critical rates of change.
The specific application of calculus here involves differentiation, which helps predict how fast a certain population parameter is changing. We used the derivative \( f'(t) \) to find the rate at which the age group segment within the population expands annually.
By applying calculus in our exercise, we see practical real-world applications such as:

• Calculating rates of growth or decline in population sub-groups.
• Modeling future scenarios to foresee potential societal challenges.
• Optimizing resources by understanding population dynamics accurately.

Rate of Change

The rate of change in a context like demographic modeling refers to how quickly or slowly a particular characteristic of a population changes over time. Using calculus, we can calculate this rate to understand a population's dynamics better.
In this problem, the rate of change at which the percentage of Americans age 55 and older is growing is acquired by differentiating the function \( f(t) \), resulting in the expression \( f'(t) \). By plugging in specific values for \( t \), such as 0 or 10, we can determine how quickly the older population segment is changing at those points.
Understanding the rate of change is vital for:

• Identifying periods of rapid demographic shifts which could impact societal structures.
• Helping governments and organizations plan for future needs and adjust resources.
• Tracking and anticipating trends in health care needs and economic support systems.

Chain Rule Differentiation

Chain rule differentiation is a calculus technique used when dealing with composite functions — functions formed by composing two or more functions. Our function \( f(t) = 10.72(0.9t+10)^{0.3} \) is such a composite function.
Applying the chain rule here involves differentiating the outer function and then multiplying it by the derivative of the inner function. The steps in our exercise showed this process:

The outer function \((0.9t+10)^{0.3}\) is differentiated first.

The inner function\( (0.9t+10) \) is then differentiated.

The chain rule is particularly beneficial when handling functions that involve layering, making it crucial for solving real-world problems like our population exercise. Mastery of this concept allows you to:

• Assess and solve problems involving complex, layered functions.
• Gain accuracy in calculations crucial across varied applied mathematics fields.
• Enhance understanding of population dynamics as intricate composite functions are broken down for analysis.