Problem 65 - Contemporary Precalculus

Question

Between 1790 and \(1860,\) the population y of the United States (in millions) in year x was given by \(y=3.9572\left(1.0299^{\circ}\right),\) where \(x=0\) corresponds to \(1790 .\)F ind the U.S. population in the given year. $$1800$$

Step-by-Step Solution

Answer

Answer: The approximate U.S. population in the year 1800 was 5.3236 million people.

Step 1: Determine the value of x that corresponds to 1800

We start by finding the value of x that corresponds to the year 1800. Since x is the number of years since 1790, we can calculate x by subtracting 1790 from 1800: x = 1800 - 1790 = 10

Step 2: Substitute the value of x into the given formula

Now that we have the value of x corresponding to the year 1800 (x=10), we can substitute it into the given formula for U.S. population: y = 3.9572 (1.0299^x) y = 3.9572 (1.0299^10)

Step 3: Solve for y to find the U.S. population in the year 1800

Finally, we compute the value of y using the formula. y = 3.9572 (1.0299^10) = 3.9572 (1.3448) ≈ 5.3236 The U.S. population in the year 1800 was approximately 5.3236 million people.

Key Concepts

Population Growth

Population growth refers to the increase in the number of individuals in a population. In the context of U.S. history, understanding how the population has grown over time can provide us with insights into economic, social, and environmental changes.
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During the late 18th and early 19th centuries, the United States experienced significant population growth. This can be attributed to factors such as high birth rates, immigration, and territorial expansion. The formula given in the exercise, \(y = 3.9572 \times (1.0299)^x\), reflects exponential growth—a common model for populations. By using this formula, we can estimate the U.S. population at different points in history.
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Understanding exponential population growth is essential because it helps us plan for resources, infrastructure, and services to accommodate increasing numbers. It also informs policies related to immigration and urban planning and has significant implications for environmental conservation.

Precalculus

Precalculus is a course that covers mathematical concepts that prepare students for calculus. It includes a variety of topics such as algebra, trigonometry, and functions, which are foundational for understanding more advanced mathematics.
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In the problem relating to U.S. population growth from 1790 to 1860, precalculus plays a crucial role in enabling us to interpret and manipulate exponential functions. Students learn how to evaluate and understand these functions, which often model real-world scenarios like population growth.
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The steps used in the original solution involve basic arithmetic and substitution, key skills in precalculus. Identifying the number of years between 1790 and 1800 as 10 (solving for \(x\)) is an example of using algebraic thinking to transform a real-world year into a useful mathematical input. This approach helps build a bridge between abstract mathematical concepts and their practical applications.

Exponential Functions

Exponential functions are mathematical models where a quantity grows at a consistent rate relative to its current value. They are expressed in the form \(y = a \cdot b^x\), where \(a\) is the initial amount, \(b\) is the growth factor, and \(x\) is the time interval.
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In the context of the original exercise, the formula \(y = 3.9572 \times (1.0299)^x\) is an example of an exponential function. Here, \(3.9572\) represents the initial population in 1790, \(1.0299\) is the annual growth factor, and \(x\) represents the number of years since 1790.
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Exponential growth like this is characterized by repeated multiplication, leading to rapid increases. This model helps illustrate why populations, and other quantities, can grow quickly under consistent conditions. Understanding these functions is also crucial in fields such as biology, economics, and finance, where exponential growth models are frequently utilized.