Problem 8 - Applied Calculus

Question

World population is approximately \(P=6.4(1.0126)^{t}\), with \(P\) in billions and \(t\) in years since 2004 . (a) What is the yearly percent rate of growth of the world population? (b) What was the world population in \(2004 ?\) What does this model predict for the world population in 2010 ? (c) Use part (b) to find the average rate of change of the world population between 2004 and 2010 .

Step-by-Step Solution

Answer

(a) The growth rate is 1.26% per year. (b) In 2004: 6.4 billion; in 2010: 6.89 billion. (c) The average rate of change is 0.0821 billion per year.

Step 1: Identify Growth Rate Formula

The given equation is \( P = 6.4(1.0126)^t \). This is an exponential growth model where the base of the exponent (1.0126) represents the growth factor each year. The growth rate \( r \) can be derived from the equation \( 1 + r = 1.0126 \).

Step 2: Calculate Yearly Percent Rate of Growth

Subtract 1 from the growth factor to find the growth rate: \( r = 1.0126 - 1 = 0.0126 \). To convert this to a percentage, multiply by 100: \( 0.0126 \times 100 = 1.26\% \).

Step 3: Determine World Population in 2004

Given that \( t = 0 \) in 2004, substitute \( t = 0 \) into the population formula: \( P = 6.4(1.0126)^0 = 6.4 \). So, the population in 2004 was 6.4 billion.

Step 4: Calculate World Population in 2010

Substitute \( t = 6 \) (2010, which is 6 years after 2004) into the model: \( P = 6.4(1.0126)^6 \). Calculate the expression: \( P \approx 6.4 \times 1.077 \approx 6.8928 \). So, the predicted population in 2010 is approximately 6.89 billion.

Step 5: Calculate Average Rate of Change from 2004 to 2010

The average rate of change is given by the formula \( \frac{P(2010) - P(2004)}{t_{2010} - t_{2004}} \). Substitute \( P(2010) = 6.8928 \) and \( P(2004) = 6.4 \) using \( t_{2010} = 6 \) and \( t_{2004} = 0 \): \( \frac{6.8928 - 6.4}{6 - 0} = \frac{0.4928}{6} \approx 0.0821 \). Thus, the average rate of change is approximately 0.0821 billion per year.

Key Concepts

World Population

The world population is an important measure that reflects the total number of people living on Earth. This figure doesn't remain static; instead, it changes over time due to births, deaths, and migration. Since the year 2004, the world population model is represented as an exponential growth equation: \[ P = 6.4(1.0126)^t \] Here, \( P \) describes the world's population in billions, and \( t \) is the number of years since 2004. Exponential growth means that the quantity grows by a constant percentage each year. It's crucial for understanding global challenges, resource allocation, and policy planning. Let's break down how this model works further in practice and predictions.

• The base population at the starting point (2004) was 6.4 billion.
• The model predicts an increase based on its growth factor each year.
• Predicted population data helps visualize potential future scenarios for planning and sustainability efforts.

By analyzing models like this, we gain insights into future trends and the sustainability of our planet's resources.

Average Rate of Change

The average rate of change provides a way to analyze how a quantity changes over a specific period of time. It's like finding the average speed if you know the distance covered over a certain duration. In the case of world population, the average rate of change between two points helps us understand overall growth, not just yearly increments. Given the world population in 2004 and 2010 using the model, we calculate this rate as follows:Formula: \[ \frac{P(2010) - P(2004)}{t_{2010} - t_{2004}} \]- Here, \(P(2010)\) was approximately 6.89 billion, and \(P(2004)\) was 6.4 billion.- The time interval \(t\) is from 2004 to 2010, which is 6 years.- Thus, the calculation gives us \( \frac{6.8928 - 6.4}{6} \approx 0.0821 \) billion per year.This result helps to understand the underlying trend of population growth over these years. It's a way to see the broader picture beyond individual yearly data and helps in assessing whether growth is accelerating or decelerating.

Yearly Percent Rate of Growth

Yearly percent rate of growth is a crucial concept that explains how much the population increases as a percentage of the previous year’s population. To find this rate, we look at the exponential model's growth factor, which, in this case, is 1.0126. This implies each year's population increases by a certain percentage over the past year's population. To determine this percentage:1. Notice that the growth factor is given by \(1 + r\), where \(r\) is the growth rate.2. From \(1.0126 = 1 + r\), we deduce that \(r = 0.0126\).3. Convert \(r\) into a percentage by multiplying by 100, resulting in a 1.26% growth per year.

• This rate tells us how quickly the world population is expected to grow annually.
• Understanding this percentage helps in anticipating resource needs and managing future population challenges.

This growth rate is not just a number but an indicator of how dynamic population trends are, guiding policy and resource management in various sectors worldwide.