/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 In 2004 , the world's population... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 30

In 2004 , the world's population was \(6.4\) billion, and the population was projected to reach \(8.5\) billion by the year 2030 . What annual growth rate is projected?

Short Answer

Expert verified
The projected annual growth rate is approximately 1.14%.

Step by step solution

01

Understanding Exponential Growth

To find the annual growth rate, we need to use the exponential growth model since population grows exponentially. The formula for exponential growth is given by \[ P(t) = P_0 (1 + r)^t \] where \( P(t) \) is the future population, \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the time in years. Our task is to find \( r \).
02

Assigning Known Values

Let's start by assigning the known values to our formula. We have the initial population in 2004, \( P_0 = 6.4 \) billion, and the projected population in 2030, \( P(t) = 8.5 \) billion. The time \( t \) is the difference between 2030 and 2004, so \( t = 26 \) years.
03

Setting up the Equation

Substitute the known values into the exponential growth formula: \[ 8.5 = 6.4 (1 + r)^{26} \]. Our task now is to solve for \( r \), the annual growth rate.
04

Isolating the Growth Factor

First, divide both sides of the equation by 6.4 to isolate the growth factor: \[ \frac{8.5}{6.4} = (1 + r)^{26} \]. Calculate \( \frac{8.5}{6.4} \) to get approximately 1.328125.
05

Solving for the Growth Rate

To solve for \( r \), take the 26th root of 1.328125: \[ 1 + r = (1.328125)^{\frac{1}{26}} \]. Compute \( (1.328125)^{\frac{1}{26}} \) which gives approximately 1.0114.
06

Finding the Growth Rate

Now solve for \( r \): \[ r = 1.0114 - 1 \] which simplifies to \( r \approx 0.0114 \).
07

Converting to Percentage

To express \( r \) as a percentage, multiply by 100: \[ 0.0114 \times 100 = 1.14\% \]. The projected annual growth rate is approximately 1.14%.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Growth
Population growth refers to the increase in the number of individuals in a population over a specified time period. It is a fundamental concept in demography and ecology. Understanding how populations grow is essential for planning and sustainability, as it affects various aspects of society such as resources, environment, and infrastructure.
Factors influencing population growth include:
  • Birth rates
  • Death rates
  • Immigration
  • Emigration
When a population grows without any restrictions, it can lead to exponential growth. This means the population size increases rapidly over time, as each generation adds more individuals than the previous one.
Hence, predicting future population sizes accurately is crucial for anticipating future needs and challenges.
Annual Growth Rate
The annual growth rate is a measure that allows us to understand how fast a population is increasing within a year. It is usually expressed as a percentage. Calculating this rate helps in assessing the pace of change in population size, which in turn can inform policy decisions and economic planning.
To find the annual growth rate, it is often necessary to use models like the exponential growth model. Once determined, this rate offers insights into whether population controls or supportive measures are needed.
The annual growth rate can be derived from the equation:
  • By using the future population size, initial population size, and the time period in years.
Exponential Growth Model
The exponential growth model is a mathematical formula used to predict population growth. It assumes that the conditions鈥攕uch as resources and space鈥攁re unlimited, allowing the population to grow continuously at a constant rate. This model is particularly useful in contexts where growth isn't significantly suppressed by external factors.
The key components of the model are:
  • \( P(t) \), the population size at a future time
  • \( P_0 \), the initial population size
  • \( r \), the growth rate as a decimal
  • \( t \), the time in years
The formula is expressed as \( P(t) = P_0 (1 + r)^t \). Each variable can impact the overall prediction of future population. By plugging the values into the formula, one can find projections for different time periods. The exponential model is highly beneficial for estimating population trends, but it should be interpreted cautiously in environments where resources or other factors may limit growth realistically.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The DuBois formula relates a person's surface area \(s\), in \(\mathrm{m}^{2}\), to weight \(w\), in \(\mathrm{kg}\), and height \(h\), in \(\mathrm{cm}\), by $$ s=0.01 w^{0.25} h^{0.75} $$ (a) What is the surface area of a person who weighs \(65 \mathrm{~kg}\) and is \(160 \mathrm{~cm}\) tall? (b) What is the weight of a person whose height is \(180 \mathrm{~cm}\) and who has a surface area of \(1.5 \mathrm{~m}^{2}\) ? (c) For people of fixed weight \(70 \mathrm{~kg}\), solve for \(h\) as a function of \(s\). Simplify your answer.

Simplify the quantities using \(m(z)=z^{2}\). $$m(z+h)-m(z-h)$$

Sketch graphs of the functions. What are their amplitudes and periods? $$y=-3 \sin 2 \theta$$

You are buying a car that comes with a one-year warranty and are considering whether to purchase an extended warranty for $$\$ 375 .$$ The extended warranty covers the two years immediately after the one-year warranty expires. You estimate that the yearly expenses that would have been covered by the extended warranty are $$\$ 150$$ at the end of the first year of the extension and $$\$ 250$$ at the end of the second year of the extension. The interest rate is \(5 \%\) per year, compounded annually. Should you buy the extended warranty? Explain.

In 2000 , there were about 213 million vehicles (cars and trucks) and about 281 million people in the US. The number of vehicles has been growing at \(4 \%\) a year, while the population has been growing at \(1 \%\) a year. If the growth rates remain constant, when is there, on average, one vehicle per person?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.