Problem 31 - Precalculus : Building Concepts and Connections

Question

The population of Florida grew from 16.0 million in 2000 to 17.4 million in \(2004 .\) (Source: U.S. Census Bureau) (a) Find a function of the form \(P(t)=C e^{t x}\) that models the population growth. Here, \(t\) is the number of years since 2000 and \(P(t)\) is in millions. (b) Use your model to predict the population of Florida in 2010.

Step-by-Step Solution

Answer

Based on the exponential growth model found, the population of Florida for the year 2010 is estimated to be approximately: \(16e^{10(\frac{ln(\frac{17.4}{16})}{4})}\) million

Step 1: Establishing The Model

The population growth model that's given is \(P(t) = Ce^{tx}\), where 'P(t)' represents the population, 't' is the time elapsed since the year 2000, 'C' is the initial population, and 'x' is the growth rate to be determined. From the initial information, we know that when 't' is '0' (year 2000), the population 'P(t)' was '16' million, so 'C' is '16'. Then, the population 'P(t)' was '17.4' million in '4' years later (year 2004). So, the equation becomes: \(17.4 = 16e^{4x}\)

Step 2: Calculating the Growth Rate Constant

Now, the aim is to work out the value of 'x'. To find 'x', divide each side of the equation by '16'. Which gives: \(\frac{17.4}{16} = e^{4x}\) . By taking the natural logarithm (ln) on both sides: \(ln(\frac{17.4}{16}) = 4x\), the growth constant 'x' would then be: \(\frac{ln(\frac{17.4}{16})}{4}\)

Step 3: Function for Population Growth

Now, substitute the value of 'x' into the function: \(P(t) = Ce^{tx}\) to get the function for the population growth as: \(P(t) = 16e^{t(\frac{ln(\frac{17.4}{16})}{4})}\)

Step 4: Predicting The Future Population

To predict the population of Florida in 2010 (which is 't' = 10 years after 2000), substitute 't' = 10 into the growth function: \(P(10) = 16e^{10(\frac{ln(\frac{17.4}{16})}{4})}\)

Key Concepts

Population Growth

Population growth can be understood as the increase in the number of individuals within a given population. However, it is not just about numbers. Population growth often considers how quickly or slowly a population grows over time.
When we look at a population, we ask questions like:

• How many people are being born?
• How many people are dying?
• How many are moving in or out of the area?

All these factors influence growth and are part of what demographers, scientists who study populations, try to measure and predict.
For our specific example, the population of Florida grew from 16 million to 17.4 million over 4 years. Using a mathematical model helps to predict these changes into the future and understand the impact of growth. With such predictions, planners and policymakers can make informed decisions about resource allocation, infrastructure development, and other social services.
Population growth is generally represented by models, and one of the most used models is the exponential growth model, which assumes that the population grows continuously and proportionally to its current size.

Growth Rate Calculation

Calculating the growth rate is a fundamental step in understanding how populations change over time. The growth rate tells us the proportional change per time unit. Let's break down the calculation:
To find the growth rate, we use an equation derived from the exponential growth model. The general form of this model is:

• \( P(t) = Ce^{tx} \)

where:

• \( P(t) \) is the population at time \( t \).
• \( C \) is the initial population.
• \( e \) is a mathematical constant approximately equal to 2.71828.
• \( x \) is the growth rate we need to find.

By observing the given data from 2000 and 2004, we use these values directly for calculations.
Using the formula \( 17.4 = 16e^{4x} \), we rearrange to discover \( x \):1. Divide both sides by 16:\( \frac{17.4}{16} = e^{4x} \)2. Apply the natural logarithm to both sides to solve for \( x \).The resulting expression \( ln(\frac{17.4}{16}) = 4x \) allows us to find \( x \) by dividing:\( x = \frac{ln(\frac{17.4}{16})}{4} \)
This process helps make future predictions using the growth model.

Natural Logarithm

A natural logarithm is a tool used to solve exponential equations, as we've seen in the calculation of growth rates. It is denoted as \( ln \) and is the logarithm to the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828.
Natural logarithms are used extensively in real-life applications because they simplify the mathematics of exponential growth and decay, making calculations manageable.
In our exercise, the natural logarithm is used to isolate the growth rate \( x \):

• The equation \( e^{4x} = \frac{17.4}{16} \) is transformed to \( ln(\frac{17.4}{16}) = 4x \) using the properties of logarithms.
• It simplifies solving for \( x \), reducing the complexity of the exponential terms.

The ability to convert between exponential and logarithmic forms makes natural logarithms particularly useful in population modeling, as it allows us to handle the steady changes in size or other quantities smoothly and accurately.
Embracing natural logarithms ensures that solutions to real-world problems, like predicting population changes, are accessible and understandable.