Problem 37 - Applied Calculus

Question

The population, \(P\), in millions, of Nicaragua was 5.4 million in 2004 and growing at an annual rate of \(3.4 \% .\) Let \(t\) be time in years since 2004 (a) Express \(P\) as a function in the form \(P=P_{0} a^{t}\) (b) Express \(P\) as an exponential function using base \(e\) (c) Compare the annual and continuous growth rates.

Step-by-Step Solution

Answer

(a) \(P(t) = 5.4 \times 1.034^t\). (b) \(P(t) = 5.4 \times e^{0.03342t}\). (c) Annual: 3.4%, Continuous: 3.342%.

Step 1: Write the Initial Population and Growth Rate

The initial population, denoted as \( P_0 \), is 5.4 million in 2004. The annual growth rate, \( r \), is given as \( 3.4\% \), which can be expressed as a decimal: \( r = 0.034 \).

Step 2: Express Population as Exponential Function with Base 'a'

We use the formula \( P = P_0 a^t \) for exponential growth. Here, \( a \) represents the growth factor, which is calculated as \( 1 + r \). Therefore, \( a = 1 + 0.034 = 1.034 \). Thus, the function becomes: \[P(t) = 5.4 \times 1.034^t\]

Step 3: Convert to Exponential Form with Base 'e'

An exponential function with base \( e \) can be expressed as \( P = P_0 e^{kt} \), where \( k \) is the continuous growth rate. We know that \( 1.034 = e^k \). Taking the natural logarithm of both sides, \( k = \ln(1.034) \). Calculate \( k \): \[k = \ln(1.034) \approx 0.03342 \] Thus, the function becomes: \[P(t) = 5.4 \times e^{0.03342t}\]

Step 4: Comparison of Growth Rates

The annual growth rate is \( 3.4\% \), while the continuous growth rate \( k \) in decimal form is approximately \( 0.03342 \), or \( 3.342\% \) when expressed in percentage. While both rates are nearly equivalent in value, the continuous rate is slightly less, indicating a closer approximation when using continuous compounding.

Key Concepts

Population Modeling

Population modeling is an essential tool that helps us understand how populations grow over time. In the context of Nicaragua, we are examining how the population increases annually using a mathematical model. The idea is to create a function that describes the population size in any given year after 2004. This function involves certain components:

• Initial Population (\(P_0\)): This is the population count at the starting point, which is 5.4 million for Nicaragua in 2004.
• Time (\(t\)): This variable represents the number of years passed since 2004.
• Growth Factor: Reflects how much the population increases per year, based on the growth rate provided.

By applying these elements, we manage to predict future population sizes and analyze trends, which can be crucial for planning resources and understanding demographic changes.

Exponential Functions

Exponential functions are crucial in describing growth processes like the increase in population. These functions take the form of \( P = P_0 a^t \), where:

• \(P\): Represents the population at time \(t\).
• \(P_0\): Is the initial amount or value, which is 5.4 million in our example.
• \(a\): The base of the exponential, is the growth factor calculated as \( 1 + r \), where \( r \) is the annual growth rate in decimal form.
• \(t\): Denotes time in years since the initial time.

These functions describe a situation where a quantity grows by a consistent percentage over each time period. For example, if the base \( a \) is 1.034, it indicates that the population grows by 3.4% annually. Exponential functions are applicable not just to populations, but also to finance, biology, and physics, wherever regular percentage growth appears.

Continuous Growth Rate

The continuous growth rate offers a more precise way to calculate growth over time using the natural number \( e \). Instead of considering growth percent by year, the factor of constant growth allows for an instant rate of increase that compounds continuously.
When expressing population with the base \( e \), the exponential function is \( P = P_0 e^{kt} \), where:

• \( k \): Represents the continuous growth rate.
• The advantage is that this method captures real-world phenomena more accurately, since growth processes can happen at any moment, not necessarily at the end of a year.

For Nicaragua's population, we find the continuous rate \( k \) by computing \( k = \ln(1.034) \approx 0.03342 \), translating to roughly 3.342%. This is slightly lower than the annual growth rate but represents a more consistent reality where growth is happening persistently over time.