Problem 80 - Applied Calculus

Question

The 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, estimate the world population in the year: 1800

Step-by-Step Solution

Answer

Roughly 886 million.

Step 1: Identify the Value of x

First, determine the value of \( x \). The year in question is 1800. Since \( x \) is the number of years since 1700, we calculate \( x = 1800 - 1700 = 100 \).

Step 2: Substitute x into the Function

Next, substitute \( x = 100 \) into the function \( P(x) = 522(1.0053)^x \). This gives us \( P(100) = 522(1.0053)^{100} \).

Step 3: Calculate the Exponential Expression

Use a calculator to compute \((1.0053)^{100}\). The result is approximately 1.698.

Step 4: Multiply by the Initial Population

Now, multiply the result from Step 3 by the initial population, which is 522. Thus, calculate \( P(100) = 522 \times 1.698 \approx 886.356 \).

Step 5: Round to the Nearest Million

Finally, round the result from Step 4 to the nearest million. Therefore, the estimated world population in 1800 is approximately 886 million.

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. This type of function frequently models real-world situations, such as population growth, radioactive decay, and interest compounding in finance. The general form of an exponential function is \( f(x) = a \, b^x \), where:

• \(a\) represents the initial value or the starting point.
• \(b\) is the base rate of growth (if \(b>1\)) or decay (if \(0
• \(x\) is the exponent, often representing time.

For example, in our problem, the exponential function \( P(x) = 522(1.0053)^x \) represents the growth of the world population over time. Here, 522 is the initial population in millions in the year 1700, and 1.0053 is the growth factor for each year.

Population Modeling

Population modeling uses mathematical formulas to predict how a population will grow or shrink over time. This involves using functions like exponential functions to depict changes. When dealing with large populations, factors such as birth rates, death rates, and immigration affect dynamics over the years. These factors often result in an exponential growth pattern if resources are unrestricted. In our example, \( P(x) = 522(1.0053)^x \) captures world population increase from 1700 onwards.By inserting \(x = 100\) for the year 1800, we can estimate how population has evolved a century later. The growth factor \(1.0053\) means a 0.53% increase each year, illustrating how small changes compound to have a significant impact over long periods. This kind of model helps build a clear picture of potential future changes as well.

Mathematical Calculations

Mathematical calculations are at the core of using exponential functions in population modeling. These calculations involve substituting given values into the function and using a calculator to perform precise computations, especially when dealing with powers. For instance:

• Determine \(x\) by subtracting the base year from the given year—\(100\) for the year 1800 in our case.
• Calculate the exponential term \((1.0053)^{100}\), which gives about 1.698.
• Multiply by the initial population figure—\(522 \times 1.698\) results in an estimated population.

Such calculations are vital for generating accurate population estimates, which allow predictions about growth trends. Rounding off the result, the estimated population of approximately 886 million in 1800 was deduced. Accurate calculations provide a reliable foundation for these analyses, bridging initial theoretical concepts with practical predictions.