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Polynomial functions are mathematical expressions involving variables raised to whole-number exponents. In our problem, we have the function \( n(x) = -0.00082x^{3} + 0.059x^{2} + 0.183x + 34.42 \). Here, \( n(x) \) models the population of Americans aged 65 or older between the years 2000 to 2050. The variables in polynomial functions have coefficients, which are numbers that multiply the variables' powers. For example, \(-0.00082, 0.059,\) and \(0.183\) are coefficients of the terms \(x^3, x^2,\) and \(x,\) respectively. The term \(34.42\) is a constant that remains the same no matter the value of \(x\). Using polynomial functions, you can calculate future projections by substituting years into the equation. For example, to find the population in 2011, substitute \(x = 11\) (since 2011 is 11 years from 2000) to find \(n(11)\). Each of the terms is then calculated by plugging into the expression, and finally, the results are summed to get the total number of the target group, projected as 42.481 million for 2011. This shows how polynomial functions can help predict future trends based on past data.

The rate of change in the context of our polynomial function refers to how quickly the population of seniors is expected to increase or decrease each year. Calculus helps us find the rate of change using derivatives. In this problem, the derivative of the polynomial function \( n(x) \) is used to find \( n'(x) = -0.00246x^2 + 0.118x + 0.183 \). The derivative, \( n'(x) \), is interpreted as the rate at which the senior population is changing at a given year \( x \). For example, to find the rate of change in 2011, substitute \(x = 11\) into the derivative, resulting in \( n'(11) = 1.18334 \). This means that in 2011, the population of older Americans was increasing by approximately 1.18 million people per year.Similarly, substituting \(x = 30\) for 2030 into the derivative yields a change rate of 1.51 million people per year. Observing rate changes across different years helps understand aspects like growth trends and planning for future infrastructural needs. For example, knowing the rate at which the elderly population increases can potentially influence healthcare planning and social security adjustments.

Percentage change is a valuable way of expressing changes in population as a proportion of the initial amount. This is particularly useful in understanding relative growth over time. In our exercise, to find the percentage change in 2030, we use the formula:\[\text{Percentage Change} = \left(\frac{n'(30)}{n(30)}\right) \times 100\]The percentage change illustrates how much the population's size changes relative to its total in that year. By calculating \( n'(30) = 1.509 \) and \( n(30) = 70.87 \), we find the percentage by solving:\[\text{Percentage Change} = \left(\frac{1.509}{70.87}\right) \times 100 \approx 2.13\%\]This calculation shows that in 2030, the projected senior population is increasing at approximately 2.13% relative to its size at that time. Understanding percentage changes is pivotal because it provides a normalized manner to discuss changes over time, and it's especially helpful when comparing different populations or time periods. This understanding can guide government policies as they work to accommodate future demographic shifts efficiently.