91Ó°ÊÓ

In mathematics, an exponential model describes how a quantity changes over time in a consistent, multiplicative way. When you work with exponential growth or decay, the essence is captured by equations of the form \( N(t) = N_0 e^{kt} \), where \( N_0 \) is the initial quantity, \( e \) is the base of natural logarithms, and \( k \) is the growth or decay constant.
The exercise you have looks at population, which is deceasing. The given model is \( \ln N = -0.051t + 1.513 \). This is a logarithmic expression which we can convert to an exponential one. Applying properties of logarithms, the expression becomes \( N = e^{1.513} \cdot e^{-0.051t} \).
This shows that our initial population \( N_0 \) is \( e^{1.513} \) and the rate of change is \(-0.051\). Measured in thousands, this describes a city whose inhabitants decrease exponentially over time.

Population decline refers to the decrease in the number of inhabitants in a specific area over time. In the context of the given exponential model, the city’s population decreases each year by a fixed percentage. Here, the rate of decline is represented by \( e^{-0.051} \) with an approximate value of 0.950, meaning each year the population retains 95% of its previous year's inhabitants.
The percentage loss can be easily calculated as \( 1 - 0.950 = 0.05 \), indicating a 5% decrease annually. This steady decline is typical in models where natural calamities, emigration, or other social factors contribute to the reduction of people within a geographic area.
Understanding this pattern helps planners anticipate future scenarios and effectively manage resources and policies.

Functional notation allows us to express mathematical relationships in a concise way. It is like a shorthand to define a function. By using this notation, we can indicate population over time, such as \( N(t) \), where \( t \) represents the years since a reference point, in this case, since 2008.
To find the population at the start of 2011, we substitute \( t = 3 \) into \( N(t) = 4.536 \cdot e^{-0.051t} \). This specific substitution is reflective of 2011 being three years past 2008. Doing the calculation, you derive \( N(3) \approx 3.887 \), denoting a population of approximately 3,887 individuals, or in terms of the problem, 3.887 thousand.
Functional notation is vital as it gives clarity while analyzing changes at specific times.

Logarithmic functions are the inverses of exponential functions. They are extremely useful in solving equations where the variable is the exponent. In the context of this exercise, finding out when the population reaches a certain level involves using logarithms.
The original equation \( N = 4.536 \cdot e^{-0.051t} \) becomes \( 3 = 4.536 \cdot e^{-0.051t} \) when solving to determine when the population will be 3,000. To isolate \( t \), take the natural logarithm of both sides, enabling you to solve for \( t \).
Through these calculations, logarithms simplify finding solutions that are otherwise difficult with only exponential forms. It's a powerful tool in analyzing situations where initial exponential forms appear to complicate direct algebraic manipulation.