Problem 79
Question
79-80.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: 1750
Step-by-Step Solution
Verified Answer
The estimated world population in 1750 is approximately 705 million.
1Step 1: Identify the Problem
We need to find the population in the year 1750 using the given function. The function provided is \( P(x) = 522(1.0053)^x \). We first determine the value of \( x \), which is the number of years since 1700, by subtracting 1700 from 1750.
2Step 2: Calculate the Value of x
Calculate \( x \) as follows: \( x = 1750 - 1700 = 50 \). Thus, we need to calculate the world population 50 years after 1700.
3Step 3: Substitute x into the Function
Substitute \( x = 50 \) into the population function to find \( P(50) \): \( P(50) = 522(1.0053)^{50} \).
4Step 4: Evaluate the Exponential Expression
Use a calculator to evaluate \( (1.0053)^{50} \). Calculating gives approximately \( 1.3498 \).
5Step 5: Calculate the Population
Multiply the result by 522 to find the approximate population in 1750: \( P(50) = 522 \times 1.3498 \). This calculates to approximately \( 704.8 \) million.
Key Concepts
Population EstimationExponential FunctionWorld Population Calculation
Population Estimation
Population estimation is a powerful tool used by demographers and statisticians to predict the size of human populations. The underlying purpose is to provide crucial data for planning and decision-making. One common model used in population estimation is the exponential growth model. This model is ideal because it captures the essence of how populations multiply under certain conditions.
To estimate the population at a specific point in time, we need to plug the number of years since a baseline, such as the year 1700, into a given function. This allows us to calculate an approximation efficiently. The approximation involves understanding how the initial population grows over time, assuming the growth rate remains constant. For example, in the given exercise, we use the year 1700 as a baseline and measure growth over a 50-year span to estimate the population in 1750.
To estimate the population at a specific point in time, we need to plug the number of years since a baseline, such as the year 1700, into a given function. This allows us to calculate an approximation efficiently. The approximation involves understanding how the initial population grows over time, assuming the growth rate remains constant. For example, in the given exercise, we use the year 1700 as a baseline and measure growth over a 50-year span to estimate the population in 1750.
Exponential Function
An exponential function is a mathematical expression in the form of \[ P(x) = a \cdot b^x \]. Here, \( a \) represents the initial value, and \( b \) is the growth factor. Exponential functions are widely used to model various natural phenomena, including population growth. They provide a reliable way of illustrating how quantities change at a rate proportional to their current value.
In the problem provided, the function \( P(x) = 522(1.0053)^x \) is used to model world population growth from the year 1700. The initial number 522 represents the world population in millions at the start, and each increment of 1 in \( x \) represents one year, adding a factor of \( 1.0053 \) to the population.
In the problem provided, the function \( P(x) = 522(1.0053)^x \) is used to model world population growth from the year 1700. The initial number 522 represents the world population in millions at the start, and each increment of 1 in \( x \) represents one year, adding a factor of \( 1.0053 \) to the population.
- Rising Exponent: As \( x \) increases, \( b^x \) grows exponentially, leading to rapid increases in population size.
- The factor \( 1.0053 \): This suggests a small but consistent growth rate which compiles over years to yield larger changes.
World Population Calculation
The calculation of world population using an exponential model involves substituting a specific year's value into the growth function. Let's break down the steps taken in our exercise. First, it's crucial to determine the correct value for \( x \) — the number of years passed since the initial year 1700. By doing so, we are fixating the appropriate growth term for the period in question.
For 1750, we calculated \( x = 50 \), which means we need to evaluate \( 522(1.0053)^{50} \). This suggests that over 50 years, the exponent in the function provides the multiplier effect of the population's growth rate.
By calculating \( (1.0053)^{50} \) radians using a calculator, we found it to be approximately 1.3498. Multiplying this result by 522 gives us the estimated population, which is approximately 704.8 million. This step illustrates how exponential growth, while initially slow, becomes significantly impactful over several decades.
For 1750, we calculated \( x = 50 \), which means we need to evaluate \( 522(1.0053)^{50} \). This suggests that over 50 years, the exponent in the function provides the multiplier effect of the population's growth rate.
By calculating \( (1.0053)^{50} \) radians using a calculator, we found it to be approximately 1.3498. Multiplying this result by 522 gives us the estimated population, which is approximately 704.8 million. This step illustrates how exponential growth, while initially slow, becomes significantly impactful over several decades.
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