Problem 81 - Algebra and Trigonometry Real Mathematics, Real People

Question

The projected populations of California for the years 2020 through 2060 can be modeled by \(P=36.308 e^{0.0065 t},\) where \(P\) is the population (in millions) and \(t\) is the time (in years), with \(t=20\) corresponding to \(2020 .\) (Source: California Department of Finance) (a) Use a graphing utility to graph the function for the years 2020 through 2060 (b) Use the table feature of the graphing utility to create a table of values for the same time period as in part (a). (c) According to the model, in what year will the population of California exceed 51 million?

Step-by-Step Solution

Answer

The exact year will depend on the results from your calculations in Step 3. However, by plotting the function and creating a table as indicated, you will be able to estimate the year in which the population of California will exceed 51 million according to the given model.

Step 1: Graph the function

First, you need to graph the function. The function in this case is \(P=36.308 e^{0.0065 t}\), where \(P\) is the population and \(t\) is the time. This will give you a visual representation of the growth trend over time. Remember that \(t=20\) corresponds to 2020. To do this, set up the graph with an x-axis that represents time \(t\) in years, and a y-axis that represents population \(P\). The graph reveals an exponential trend which helps visualize the rapid growth.

Step 2: Create a table of values

Using the table feature of a graphing calculator, input the given function: \(P=36.308 e^{0.0065 t}\). Then, set values for \(t\) ranging from 20 to 60, which corresponds to the years 2020 to 2060 in this case. The corresponding values for \(P\) will display the projected population for each year in millions.

Step 3: Determine when the population will exceed 51 million

In order to determine when the population exceeds 51 million, solve the given function for \(t\) when \(P > 51\). Use a graphing calculator or another numerical method to find the value of \(t\) that makes this true. The corresponding value for \(t\) will give the year when the population of California will exceed 51 million.

Key Concepts

Graphing Utility

When tackling exponential growth models like the one provided for California's population, a graphing utility becomes indispensable. A graphing utility is a technological tool, often physical like a calculator or software, that helps visualize functions and equations.

To graph the function given in the problem, input the equation \( P = 36.308 e^{0.0065 t} \) into the graphing utility. This function models population growth, where \( P \) is in millions.

• Set the horizontal axis or x-axis to represent time in years (with \( t = 20 \) being 2020).
• The vertical axis or y-axis will show the population in millions.

By graphing this function, you will see an upward trend indicating exponential growth. This visual aid helps understand how rapidly the population might increase over the years.

Population Projection

Population projections are essential for planning and understanding potential future scenarios. In the exercise, the formula \( P = 36.308 e^{0.0065 t} \) gives the population of California in millions, with \( t = 20 \) corresponding to 2020. This formula is based on the exponential growth model, which assumes the population grows at a constant percentage rate annually.

Understanding this projection helps government and organizations plan for the future. It influences infrastructure, healthcare, and educational facilities planning.

• Although projections provide estimates, real-life factors like migration, economics, and policy changes can affect outcomes.
• It's beneficial for policymakers to regularly update these projections to reflect current data and trends.

Accurate and current population projections assist in effective strategic planning.

Table of Values

Creating a table of values provides a concrete way to see how numbers change over time. By using the table feature on your graphing utility, you can input the function \( P = 36.308 e^{0.0065 t} \) and generate specific population figures.

Follow these steps:

• Enter the function into the calculator.
• Select the table mode.
• Set \( t \) values to range between 20 and 60 (these correspond to the years 2020 to 2060).
• Read off the corresponding \( P \) values, indicating the model's projected population in millions for each year.

Viewing these numbers in a table can make it easier to identify trends or key points, such as when the population exceeds critical thresholds.

California Population

The exercise's context revolves around predicting the future population of California. California, being one of the most populous states, has a significant impact on national and economic planning.

Using the model \( P = 36.308 e^{0.0065 t} \), it's possible to project the population beyond current figures to plan for future needs.

• One important projection goal is determining when the population will exceed 51 million.
• Set \( P = 51 \) in the equation and solve for \( t \), using your graphing utility or solving methods to find when this threshold is crossed.

Such projections are crucial for urban planning, resource allocation, and policy development to ensure sustainable growth and quality of life.