Problem 41 - Algebra 2

Question

In 2005 , the world's population was about 6.5 billion. If the world's population continues to grow at a constant rate, the future population \(P,\) in billions, can be predicted by \(P=6.5 e^{0.02 t},\) where \(t\) is the time in years since 2005. According to this model, what will the world’s population be in 2015?

Step-by-Step Solution

Answer

The world's population in 2015 is approximately 7.939 billion.

Step 1: Understand the problem

We need to determine what the world's population will be in the year 2015 using the provided exponential model equation for population growth.

Step 2: Identify given values

The equation given is \( P = 6.5e^{0.02t} \). Here, we know that \( t \) is the number of years since 2005 and emailgt;2005 and \( e \) is the mathematical constant eTt approximately 2.718.

Step 3: Calculate time difference

To find \( t \) for the year 2015, we subtract 2005 from 2015, resulting in \( t = 2015 - 2005 = 10 \).

Step 4: Substitute the time into the equation

We substitute \( t = 10 \) into the population equation: \( P = 6.5e^{0.02 \times 10} \).

Step 5: Perform the calculation

Calculate the exponent \( 0.02 \times 10 = 0.2 \). Then compute \( e^{0.2} \) using a calculator, which is approximately 1.2214. Multiply by 6.5: \( P = 6.5 \times 1.2214 \approx 7.939 \).

Step 6: Interpret the result

Therefore, in 2015, the world's population is estimated to be approximately 7.939 billion.

Key Concepts

World Population

Understanding world population growth is crucial to various sectors such as economics, environmental science, and public health. The population of the world has been changing rapidly over the past few decades. It's important to comprehend these changes not just in numbers but in what they represent for global food supply, housing, healthcare, and resources.

In simple terms, the world population is the total number of humans currently living on Earth. From the beginning of organized human settlement, the population has gone through phases of growth and decline. Recent centuries have seen unprecedented growth due to advancements in agriculture, sanitation, and medical care.

Tracking population data helps policymakers make informed decisions. They can plan for future growth by designing effective policies that cater to an increasing number of people. These include infrastructure developments, urban planning, and healthcare services. Therefore, understanding how population changes over time is not just about numbers; it's about sustainability and preparing for the future.

Mathematical Modeling

Mathematical modeling is a powerful tool used to represent real-world phenomena through mathematical language. In the context of world population, it provides a way to predict and study demographic changes, enabling us to visualize future scenarios and plan accordingly.

Population modeling involves applying mathematical equations to simulate the behavior of population dynamics. For instance, we can model the population growth with equations like the one given in our exercise: \[ P = 6.5 e^{0.02t} \]

This model uses the constant rate of growth to predict the future population. By using mathematical models:

• We can quickly assess the impact of different growth rates.
• They help in evaluating the effects of new policies or changes in birth and death rates.
• They provide insight into potential challenges or opportunities arising from population changes.

Using models, we can test hypotheses and explore the outcomes of different scenarios without waiting to observe real-time changes. This is invaluable for proactive policy-making and strategic planning.

Exponential Functions

Exponential functions are a key mathematical concept when discussing population growth and other processes occurring continuously and at increasing rates. These functions are of the form \( f(x) = a e^{bx} \), where \( e \) is Euler's number, approximately equal to 2.718.

Exponential growth, specifically, describes a process where the rate of change is proportional to the current value, leading to growth that accelerates over time. In the world population context, when population growth follows an exponential trend, it means the larger the population, the faster it grows.

In our example:\[ P = 6.5 e^{0.02t} \]

• "6.5" is the initial population in billions in 2005.
• "0.02" represents the growth rate, indicating a 2% growth per year.

Exponential growth models are powerful as they can illustrate a huge range of natural phenomena. However, it's crucial to remember that real-world limits might eventually affect growth, and so models might need adjustment over long periods. Understanding exponential functions allows us to analyze and predict how populations can potentially surge if unchecked.