Problem 42 - Precalculus with Limits

Question

The populations \(P\) (in thousands) of a city from 2000 through 2010 can be modeled by \(P=\frac{2632}{1+0.083 e^{0.050 t}}\) where \(t\) represents the year, with \(t=0\) corresponding to 2000 (a) Use the model to find the populations of the city in the years \(2000,2005,\) and 2010 . (b) Use a graphing utility to graph the function. (c) Use the graph to determine the year in which the population will reach 2.2 million. (d) Confirm your answer to part (c) algebraically.

Step-by-Step Solution

Answer

To solve this problem, the populations for the years 2000, 2005, and 2010 should be calculated by substituting \(t\) as 0, 5, and 10 respectively in the given equation \(P = \frac{2632}{1+0.083 e^{0.050 t}}\). Then, by plotting the function, the year in which the population reaches 2.2 million can be approximated. Lastly, by setting \(P\) equal to 2200 in the equation and solving for \(t\), the year can be confirmed algebraically.

Step 1: Calculate population for given years

For year 2000, \(t=0\), so substituting in the given equation, we get \(P=\frac{2632}{1+0.083 e^{0.050 * 0}} = 2632 \) thousands. For year 2005, \(t=5\), so, \(P= \frac{2632}{1+0.083 e^{0.050 * 5}}\) thousands. For year 2010, \(t=10\), so, \(P= \frac{2632}{1+0.083 e^{0.050 * 10}}\) thousands.

Step 2: Draw the graph

For this step, we need a graphing utility such as graphing calculator or computer software. The function\(P= \frac{2632}{1+0.083 e^{0.050 t}}\) should be plotted along with year \(t\) on x-axis and population \(P\) on y-axis.

Step 3: Determine the year when the population reaches 2.2 million

By visually inspecting the graph from step 2, locate the point where \(P\) crosses 2200 (since, in the equation \(P\) is in thousands), and find the corresponding year \(t\).

Step 4: Confirm the year algebraically

To confirm the year algebraically, set \(P\) in the equation equal to 2200 \(P = 2200\) (since 2.2 million equals 2200 thousands), and solve for \(t\) which will give the year.

Key Concepts

Exponential Growth

Exponential growth is a fascinating concept in population modeling. When a population grows exponentially, it increases at a rate proportional to its current size. This results in the population growing faster as it becomes larger. In mathematical modeling, we often use functions involving exponents to describe such situations.

In the given equation for the city's population, \( P = \frac{2632}{1+0.083 e^{0.050t}} \), we see an exponential term, \( e^{0.050t} \). Though this looks complex, it's a common form where \( e \) is the base of the natural logarithms, roughly equal to 2.71828.

**Why Use Exponential Models?**

• Exponential models capture rapid growth patterns effectively.
• They are often used when resources are unlimited, like during the initial phase of growth.
• They provide insights into how quickly a population can reach large sizes.

Understanding exponential growth gives us a powerful tool to predict future population sizes and make informed decisions about resource management.

Graphing Utility

A graphing utility, such as a graphing calculator or computer software, is invaluable in visualizing mathematical functions like population growth. Graphing allows us to see the function's behavior over time, identify trends, and make predictions.

To graph the population function, \( P = \frac{2632}{1+0.083 e^{0.050t}} \), you would set up a coordinate system with the year \( t \) on the x-axis and the population \( P \) (in thousands) on the y-axis.

**Steps for Using a Graphing Utility**

• Input the function into the graphing tool.
• Adjust the window settings to cover the desired range, for example, years 2000 to 2010.
• Examine intersections, peaks, and trends directly from the graph.

By using a graph, you can visually estimate the year when the population reaches milestones, like 2.2 million (2200 thousands). This visual method complements algebraic solutions by confirming and providing a quick overview of the results.

Algebraic Verification

Algebraic verification is the process of confirming results through calculations. By doing so, we ensure that our interpretations—especially those from graphical methods—are accurate. In our model, we used algebraic verification to determine when the population reaches a specific size.

For the population model \( P = \frac{2632}{1+0.083 e^{0.050t}} \), suppose we want the population to be 2.2 million, or 2200 in thousands. This requires solving the equation:\[2200 = \frac{2632}{1 + 0.083 e^{0.050t}}\]By isolating \( t \), we find:

• First, multiply both sides by \(1 + 0.083 e^{0.050t} \) and divide by 2200.
• This simplifies to finding \( t = \frac{\ln(\frac{2632}{2200} - 1)/0.083}{0.050}\) after further steps.
• Finally, calculate \( t \), which gives the exact year when the population reaches 2.2 million.

Algebraic verification strengthens our ability to predict future events accurately and reassures that our graphing or estimates are correct. It ensures a clear connection between theoretical models and real-world applications.