Problem 22 - Calculus

Question

In 2010 , the world's population reached 6.91 billion and was increasing at a rate of \(1.1 \%\) per year. Assume that this growth rate remains constant. (In fact, the growth rate has decreased since \(1987 .)\) (a) Write a formula for the world population (in billions) as a function of the number of years since 2010 (b) Estimate the population of the world in the year 2020 (c) Sketch world population as a function of years since 2010\. Use the graph to estimate the doubling time of the population of the world.

Step-by-Step Solution

Answer

Formula: \( P(t) = 6.91 \cdot (1.011)^t \); 2020 population estimate: 7.71 billion; doubling time: ~64 years.

Step 1: Understand the Exponential Growth Formula

To model the world population growth, we can use the exponential growth formula: \( P(t) = P_0 \cdot (1 + r)^t \). Here, \( P_0 \) is the initial population (6.91 billion in 2010), \( r \) is the growth rate per year (1.1% or 0.011 as a decimal), and \( t \) is the number of years since 2010.

Step 2: Write the Formula for World Population

Using the information given, we can write the formula for the world population as a function of years since 2010 as follows: \( P(t) = 6.91 \cdot (1 + 0.011)^t \). This equation will allow us to calculate the estimated population for any year \( t \) years after 2010.

Step 3: Calculate the Population in 2020

To estimate the population in 2020, we set \( t = 10 \) because 2020 is 10 years after 2010. Substitute \( t = 10 \) into the formula: \( P(10) = 6.91 \cdot (1.011)^{10} \). Calculating this gives: \( P(10) \approx 6.91 \cdot 1.116 \approx 7.71 \) billion.

Step 4: Sketch the Population Growth

The function \( P(t) = 6.91 \cdot (1.011)^t \) is an exponential curve that increases over time. To sketch this graph, plot several key points: \((0, 6.91)\), \((10, 7.71)\), and a few others. The graph should start at 6.91 billion, and by 2020, reach around 7.71 billion, showing a smooth exponential rise.

Step 5: Estimate the Doubling Time

Using the Rule of 70, the doubling time can be estimated by dividing 70 by the percentage growth rate. So, the doubling time is \( \frac{70}{1.1} \approx 63.64 \) years. From the graph, confirm that it takes around 64 years for the population to double from 6.91 billion to approximately 13.82 billion.

Key Concepts

Population Growth

Population growth is a fascinating and fundamental concept in understanding how the world changes over time. It refers to the increase in the number of people in a population, which can be both natural and influenced by various factors. In our exercise, we consider a scenario where the global population grows at a constant rate.
In 2010, the world population was 6.91 billion, and it's assumed to grow at an annual rate of 1.1%. Even a slight change in the growth rate can significantly impact future population numbers.

• A constant growth rate means the percentage increase from year to year remains the same. This makes calculations predictable and manageable.
• Population growth can be affected by birth rates, death rates, and migration. In our simplified model, we only consider the growth rate as a constant factor.

In this scenario, understanding how the population growth behaves over time helps in planning for resources, infrastructure, and services to meet future needs.

Exponential Functions

Exponential functions are mathematical expressions used to model situations where growth accelerates over time. They are characterized by a constant rate of growth, which in this case, is an increase by 1.1% per year. An exponential growth formula is written as:
\[ P(t) = P_0 \cdot (1 + r)^t \]
Where:

• \(P(t)\) is the population after \(t\) years.
• \(P_0\) is the initial population.
• \(r\) is the growth rate (expressed as a decimal).
• \(t\) is the time in years.

For the world's population starting at 6.91 billion in 2010, our formula becomes:
\[ P(t) = 6.91 \cdot (1 + 0.011)^t \]
This formula helps us predict populations in future years. Exponential growth leads to dramatic increases over time, as each year the growth builds on the previous year's population.

Doubling Time

Doubling time is an intuitive way to understand how quickly a population, or anything that follows exponential growth, grows to twice its size. It's calculated using a handy rule: the Rule of 70.
The Rule of 70 states:
\[ \text{Doubling Time} = \frac{70}{\text{growth rate as a percentage}} \]
For our population growth exercise, the growth rate is 1.1%. Thus, the doubling time is:

• \(\frac{70}{1.1} \approx 63.64\) years.

This means it will take approximately 64 years for the world population to double from its initial size of 6.91 billion to about 13.82 billion. Knowing the doubling time is crucial for understanding long-term impacts and planning, allowing us to anticipate future demands and challenges.