Problem 39 - Applied Calculus

Question

The population of the world can be represented by \(P=\) \(7(1.0115)^{t},\) where \(P\) is in billions of people and \(t\) is years since \(2012 .\) Find a formula for the population of the world using a continuous growth rate.

Step-by-Step Solution

Answer

The continuous growth formula is \(P = 7e^{0.0114t}\).

Step 1: Understand the given formula

The formula given is \(P = 7(1.0115)^t\). Here, \(P\) is the population in billions, \(7\) is the starting population in 2012, and \(1.0115\) is the growth factor per year. \(t\) represents the number of years since 2012.

Step 2: Convert to continuous growth formula format

The continuous growth formula is \(P = P_0 e^{rt}\), where \(P_0\) is the initial amount, \(r\) is the continuous growth rate, and \(e\) is the base of the natural logarithm. We need to find the equivalent of the growth factor \(1.0115\) in terms of \(e^{rt}\).

Step 3: Set the discrete and continuous formulas equal

To find the equivalent continuous rate, set \(1.0115\) equal to \(e^r\). The goal is to solve for \(r\).

Step 4: Solve for continuous growth rate \(r\)

Solve \(1.0115 = e^r\) by taking the natural logarithm of both sides: \(\ln(1.0115) = r\). Calculate \(r \approx 0.0114\) as the continuous growth rate.

Step 5: Write the formula using continuous growth rate

Substitute \(r\) into the continuous growth formula: \(P = 7e^{0.0114t}\). This is the formula for the population of the world using a continuous growth rate.

Key Concepts

Population Growth

Population growth is the change in the number of individuals in a population over time. In the context of world population, this growth is influenced by birth rates, death rates, and migration patterns. An important aspect of modeling population growth is its ability to provide insight into future population sizes and help with planning for resources, such as food and housing. When modeling population growth, we often start with an initial population size and then apply a growth factor that accounts for the increase or decrease in size over each time period. This growth factor can be influenced by economic conditions, health care, and societal factors. A common model used in population studies is the exponential growth model, where population grows by a consistent percentage over each time period. It is crucial to accurately model population growth to forecast future needs and to make informed decisions about resource allocation, urban planning, and environmental considerations.

Exponential Growth

Exponential growth occurs when a quantity, such as population, increases at a consistent percentage rate over equal intervals of time. This type of growth can be described by mathematical models such as the discrete exponential growth equation: \[P = P_0(1 + r)^t\] In this equation, \(P_0\) is the initial quantity, \(r\) is the growth rate, and \(t\) is the period of growth. A key characteristic of exponential growth is its rapid increase, meaning the larger the population, the faster it grows. This is why the exponential growth model is used when considering unbridled population increases under optimal conditions. While exponential growth is a powerful model for understanding dynamics such as early human population increases, it doesn't account for factors like resource limitations. Constant exponential growth isn't sustainable indefinitely due to eventual limitations in resources and space, which can eventually slow down growth.

Natural Logarithm

The concept of the natural logarithm, denoted as \(\ln\), is vital in converting exponential growth models into continuous growth models. The natural logarithm is the inverse function of the exponential function using the base \(e\), where \(e\) is approximately equal to 2.71828.In the original exercise, we used the natural logarithm to convert the discrete growth factor into a continuous rate. By equating the discrete growth factor, 1.0115, to \(e^r\), we can solve for the continuous growth rate \(r\) using \(\ln\). This is achieved through the equation: \[\ln(1.0115) = r\]The natural logarithm helps bridge the discrete model, which shows increments over specific intervals, with a continuous model, which assumes growth occurs uniformly over time. Understanding and utilizing natural logarithms allow for more flexible modeling of growth scenarios, providing more accurate predictions for scenarios like human population growth.