Problem 53 - Algebra 2

Question

For Exercises \(53-55,\) use the following information. Every ten years, the Bureau of the Census counts the number of people living in the United States. In \(1790,\) the population of the U.S. was 3.93 million. By \(1800,\) this number had grown to 5.31 million. Write an exponential function that could be used to model the U.S. population \(y\) in millions for 1790 to \(1800 .\) Write the equation in terms of \(x,\) the number of decades \(x\) since \(1790 .\)

Step-by-Step Solution

Answer

The function is \( y = 3.93 \cdot 1.351^x \).

Step 1: Understand the Exponential Model Formula

The formula for an exponential growth function is \( y = a \cdot b^x \), where \( a \) is the initial amount, \( b \) is the growth factor, and \( x \) is the number of time periods elapsed (in this case, decades from 1790).

Step 2: Determine Initial Value \(a\)

The initial population at 1790 is given as 3.93 million. Thus, \( a = 3.93 \).

Step 3: Define the Variable \(x\)

The variable \( x \) represents the number of decades since 1790. For example, \( x = 1 \) corresponds to 1800, 10 years (or 1 decade) after 1790.

Step 4: Calculate the Growth Factor \(b\)

To find \( b \), use the information that the population grew to 5.31 million by 1800, which is 1 decade after 1790. Therefore, the equation will be \( 5.31 = 3.93 \cdot b^1 \). Solve for \( b \) by dividing both sides by 3.93: \( b = \frac{5.31}{3.93} \approx 1.351 \).

Step 5: Construct the Exponential Function

With \( a = 3.93 \) and \( b \approx 1.351 \), the exponential function modeling the population is \( y = 3.93 \cdot 1.351^x \).

Key Concepts

Census Data

Census data refers to the comprehensive counting of a nation's population, conducted regularly by government agencies, such as the Bureau of the Census in the United States. This data collection helps governments and researchers to understand demographic changes and trends over time. A census provides a snapshot of how many people live in a country at a certain time, along with their characteristics such as age, gender, and ethnicity.

Understanding census data is crucial for making informed decisions in policy-making, urban planning, and resource allocation.

• Census data allows cities to plan for housing and infrastructure needs.
• It helps allocate federal funding and representation in government.
• Population trends can be tracked and modeled mathematically.

In the context of population growth, census data lays the groundwork for modeling and projecting future changes in population, enabling forecasts and strategic planning.

Exponential Function

An exponential function is a mathematical expression used to model situations where growth or decay occurs at a constant percentage rate over time. In general, the formula for an exponential growth function is \( y = a \cdot b^x \), where:

• \( y \) is the final amount or population.
• \( a \) represents the initial amount or population (the value at \( x = 0 \)).
• \( b \) is the growth factor, which indicates how much the quantity multiplies per each time period.
• \( x \) stands for the number of time periods that have passed.

In cases of population growth like the U.S. population from 1790 to 1800, an exponential function offers a simple yet powerful way to capture and predict growth trends. By using a consistent growth factor \( b \), it helps us understand how small changes over time lead to significant overall impacts.

Population Modeling

Population modeling involves using mathematical formulas and statistical methods to represent and analyze population dynamics. By choosing a suitable model, like the exponential function, we can describe and forecast population changes over time, based on historical data.

For example, to model the U.S. population from 1790 to 1800, we use an exponential function with known parameters obtained from census data:

• The initial population in 1790 was 3.93 million, setting \( a = 3.93 \).
• By 1800, the population grew to 5.31 million, allowing us to calculate the growth factor \( b \approx 1.351 \).

Therefore, the exponential model \( y = 3.93 \cdot 1.351^x \) represents how the population grows over each decade \( x \).

Population modeling is invaluable for predicting future trends and making decisions about resources, urban development, and environmental impact. It provides insight into the potentially exponential nature of growth and helps assess long-term sustainability concerns.