Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Population The population \(P\) of a city is given by \(P=120,000 e^{0.016 t}\) where \(t\) represents the year, with \(t=0\) corresponding to 2000\. Sketch the graph of this equation. Use the model to predict the year in which the population of the city will reach about 180,000

Step-by-Step Solution

Answer

The sketch of the graph will be an upward curving line starting from the point (0,120000). The year that the city population will reach about 180,000 can be calculated to be approximately 16.5 years after 2000, which corresponds to the year 2017.

Step 1: Understanding the exponential growth formula

The exponential growth model is given by \(P=120,000 e^{0.016 t}\). Here, P is the population at time 't', 120,000 is the initial population at \(t = 0\) (Year 2000), e is the natural logarithm base approximately 2.71828, 0.016 is the annual growth rate, and 't' is the time in years from the year 2000.

Step 2: Sketching the graph

To sketch the graph, take 't' on x-axis and 'P' on y-axis. Start from the point (0,120000) because the population is 120,000 at year 2000. As 't' increases, 'P' will also increase because of the positive exponent. It won't be a straight line but an upward curving line (due to the nature of exponential growth).

Step 3: Predicting when the population will reach 180,000

To predict when the population will reach 180,000, simply solve the equation \(120,000 e^{0.016 t} = 180,000\). First, divide both sides by 120,000 to get \(e^{0.016 t} = 1.5\). Then, take natural log of both sides to get \(0.016t = ln(1.5)\). Finally, solve for 't' by dividing both sides by 0.016. This gives the number of years from the year 2000.

Key Concepts

Population Model

A population model is a mathematical representation used to predict the growth or decline of a population over time. In our example, the population model used is an exponential model represented by the formula \(P = 120,000 e^{0.016 t}\). This model describes how the population \(P\) of a city changes over time \(t\).In this equation:

• The initial population at the starting year (2000) is 120,000.
• The variable \(t\) represents the number of years elapsed since 2000.
• The growth factor \(e^{0.016 t}\) accounts for how the population grows exponentially.

Exponential models are often used in real-world scenarios where growth accelerates over time, such as populations, investments, or biological systems. As time progresses, the population increases at a rate proportional to its current size, leading to a rapidly growing curve as opposed to a straight line. Understanding this model helps in predicting future population sizes and recognizing patterns in growth.

Annual Growth Rate

The annual growth rate is a crucial component of understanding how populations evolve over time. In the given equation, the annual growth rate is represented by the exponent 0.016. This rate is the constant percentage increase in population size per year. Here's how it works:

• The 0.016 is derived from real-world data indicating that the population increases by roughly 1.6% each year.
• It informs us of the speed and nature of growth, making predictions about future population sizes possible.

The annual growth rate serves as a guide for estimating future population levels. With this rate, policymakers and planners can decide how to allocate resources effectively, such as housing, infrastructure, and services, to accommodate the growing population. Knowing the annual growth rate is essential for understanding the dynamics of population change.

Natural Logarithm

The natural logarithm is a fundamental concept in mathematics often used in exponential growth calculations. It is denoted as \(\ln\) and works with the natural base \(e\) (approximately 2.71828).In step 3 of solving the problem, the natural logarithm helps to isolate the variable \(t\) when predicting the future population:

• By taking the natural log of both sides, you transform the exponential equation \(e^{0.016 t} = 1.5\) into a linear equation: \(0.016 t = \ln(1.5)\).
• This step allows for easier manipulation and solution of the equation to find \(t\).

The natural logarithm serves as a powerful tool in mathematics because it can "undo" exponential functions and help solve equations where time or growth rates are unknown. In this case, \(t\) is calculated by dividing \(\ln(1.5)\) by 0.016, giving the number of years until the population reaches 180,000.