91Ó°ÊÓ

When we talk about exponential growth, understanding how to calculate the growth rate is crucial. The growth rate in an exponential model can be found using the formula: \[ P_t = P_0 \times e^{rt} \]. Here, the growth rate, denoted by \( r \), tells us how fast the population is increasing over time. To find \( r \), we'll rearrange the formula and use the provided population data.
To illustrate, consider the given data from the exercise:

• Initial population, \( P_0 \), in 1776: 2,508,000
• Population in the bicentennial year, \( P_t \), in 1976: 216,000,000
• Time period \( t \): 200 years

We start by setting up the equation: \[ 216,000,000 = 2,508,000 \times e^{200r} \]
Then, to isolate \( r \), we first divide both sides by 2,508,000: \[ \frac{216,000,000}{2,508,000} = e^{200r} \]. This simplifies to: \[ 86.14 \rightarrow e^{200r} \]. Using the natural logarithm, we have: \[ \text{ln}(86.14) = 200r \]. Finally, we solve for \( r \): \[ r = \frac{\text{ln}(86.14)}{200} \]. Using a calculator, \[ r \rightarrow 0.0223 \approx 2.23\text{%} \].
This percentage represents the annual growth rate over the 200 years.

The population growth model we're using here is the exponential growth model. This model assumes that the population grows proportionally to its current size, meaning the larger the population, the faster it grows. The general formula for this model is: \[ P_t = P_0 \times e^{rt} \], where:

• \( P_t \) is the population after time \( t \)
• \( P_0 \) is the initial population
• \( r \) is the growth rate
• \( t \) is the time period

Exponential models are useful because they can provide a straightforward way to project future growth based on historical data.
However, one must consider if the exponential model is a reasonable assumption.
Over a long period, many factors such as wars, immigration, economic changes, and government policies can significantly impact population growth.
In this exercise, given the extensive period (200 years), using an exponential growth model provides a simple and useful estimation, despite potential variations caused by these factors.

The natural logarithm, often denoted as \( \text{ln} \), is a special type of logarithm where the base is the constant \( e \) (approximately 2.71828). It is a crucial mathematical tool in solving exponential equations. Here's why it's helpful:

• It allows us to work with exponential equations more easily.
• It transforms multiplicative relationships into additive ones, simplifying calculations.

In our context, when we're given the equation: \[ 86.14 \rightarrow e^{200r} \], solving for \( r \) involves taking the natural logarithm of both sides: \[ \text{ln}(86.14) = 200r \]. By applying the \( \text{ln} \), we convert the problem into a linear equation which is much easier to handle. Finally, solving for \( r \) gives us: \[ r = \frac{\text{ln}(86.14)}{200} \], played an essential role by simplifying the equation, and giving us the precise growth rate needed for the population model.
Understanding the concept and use of the natural logarithm is fundamental when dealing with exponential growth or decay models.