Problem 80 - Brief Applied Calculus

Question

world population (in millions) since the year 1700 is approximated by the exponential function \(P(x)=522(1.0053)^{x}\), where \(x\) is the number of years since 1700 (for \(0 \leq x \leq 200\) ). Using a calculator, esti mate the world population in the year: 1800

Step-by-Step Solution

Answer

In 1800, the estimated world population was approximately 906 million.

Step 1: Identify the Value of x

The exercise asks for the population in the year 1800. First, we determine the value of \(x\) by subtracting 1700 from 1800: \(x = 1800 - 1700 = 100\). This value will be used in the exponential function.

Step 2: Substitute x in the Function

Substitute \(x = 100\) into the formula \(P(x)=522(1.0053)^{x}\). This gives us \(P(100)=522(1.0053)^{100}\).

Step 3: Calculate the Exponential Expression

Calculate \((1.0053)^{100}\) using a calculator. This will give the growth factor for the population since the year 1700.

Step 4: Multiply by Initial Population Factor

Once you have \((1.0053)^{100}\), multiply it by 522 to get the population estimate. This represents the world population in millions for the year 1800.

Step 5: Compute and Interpret the Result

Perform the calculation: \(P(100) = 522 \times (1.0053)^{100}\) using a calculator. The result will be an estimate of the world population in the year 1800.

Key Concepts

world population

The world population is a complex and dynamic measure that shows how the number of people in the world changes over time. It's influenced by various factors such as birth rates, death rates, and migration patterns. Understanding the trends in world population helps policymakers and researchers make informed decisions about resource allocation and environmental planning.

In historical studies, estimating past populations can provide insights into how human societies have evolved. The population of the world has been on the rise due to advances in healthcare, agriculture, and technology. These advances have significantly reduced mortality rates and increased the average lifespan. In our exercise, we look at a specific function to approximate the world population since the year 1700. This type of approximation helps us predict and understand the population dynamics during that period.

exponential function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. This creates rapid increases or decreases over time, showing patterns of growth or decay. For example, if we consider the expression \[ P(x) = 522(1.0053)^{x} \]this is an exponential function where the base 1.0053 represents a growth factor, and the exponent \(x\) illustrates the number of time periods. Exponential functions are powerful tools in modeling growth processes, such as populations, because they can closely mimic real-life phenomena.

Applications of exponential functions are vast and varied, often used in fields like biology, finance, physics, and environmental science. In the context of world population, they help describe how populations grow over time, accounting for compound growth patterns. These functions can also be visually represented as curve graphs, which typically illustrate how the rate of change becomes more pronounced as time progresses.

mathematical modeling

Mathematical modeling is a method used to represent and analyze real-world systems using mathematical concepts and language. It serves as a bridge between mathematical theories and practical applications. By constructing models like the exponential function given in the exercise, we can simulate and predict real-world scenarios. This particular model focuses on estimating the world population growth over decades.

The aim of mathematical modeling is to simplify complex systems into understandable equations that can describe behavior or predict outcomes. It's important to note that while models like \( P(x) = 522(1.0053)^{x}\) provide valuable insights, they may not capture every real-world variable. As such, assumptions are made to focus on the most impactful factors, and the results are approximate rather than exact.

Effective mathematical modeling involves validating these approximations against known data and adjusting the models to improve accuracy where necessary. This process not only helps in understanding past trends but also in forecasting future developments, making mathematical modeling an essential tool in research and decision-making processes.