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Chapter 5: Problem 22

Population growth in India The 1985 population estimate for India was 766 million, and the population has been growing continuously at a rate of about \(1.82 \%\) per year. Assuming that this rapid growth rate continues, estimate the population \(N(t)\) of India in the year \(2015 .\)

Short Answer

Expert verified
The estimated population of India in 2015 is approximately 1323 million.

Step by step solution

01

Identify the Given Information

We are provided with the initial population in 1985 as 766 million. The continuous growth rate is given as 1.82% per year.
02

Understand the Growth Formula

For continuous growth, we use the formula for exponential growth: \[ N(t) = N_0 imes e^{rt} \]where \(N_0\) is the initial population, \(r\) is the growth rate (in decimal), and \(t\) is the time in years.
03

Convert the Growth Rate to Decimal

The growth rate is 1.82%, which can be converted to a decimal: \[ r = \frac{1.82}{100} = 0.0182 \]
04

Calculate the Time Period

Determine the number of years between 1985 and 2015. This is \[ t = 2015 - 1985 = 30 \] years.
05

Apply Values to the Growth Formula

Substitute the known values into the exponential growth formula: \[ N(t) = 766 \times e^{0.0182 \times 30} \]
06

Solve the Exponent

Calculate the exponent:\[ 0.0182 \times 30 = 0.546 \]
07

Calculate the Exponential Term

Calculate the exponential term using a calculator:\[ e^{0.546} \approx 1.726 \]
08

Compute the Final Population

Determine the estimated population by multiplying:\[ N(30) = 766 \times 1.726 \approx 1322.516 \] million.

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Key Concepts

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

Population Growth
Population growth refers to the change in the number of individuals in a given area over a specific period. Understanding population growth is crucial as it helps governments plan resources, infrastructure, and services necessary for its people. Population growth can be positive or negative, depending on whether the population is increasing or decreasing.

There are two common types of population growth:
  • Exponential Growth: Occurs when the growth rate of the population is proportional to its size, leading to faster increases as the population grows.
  • Logarithmic Growth: Occurs when the growth slows down as the population reaches a certain point, often due to limited resources.
In our exercise, India's population is modeled using exponential growth, indicating that the population is increasing at a constant rate over time. This is typical in scenarios where a population grows without constraints, such as limits in resources or space.
Continuous Growth Rate
The continuous growth rate is a rate that expresses how quickly a population grows without interruption over time. Unlike a discrete growth rate that may apply at specific intervals, continuous growth is ongoing and never stops for a particular period.

Calculating the continuous growth rate requires understanding it as a percentage and converting it into a decimal form for calculations. For instance, a continuous growth rate of 1.82% is converted into a decimal as follows:
  • The rate in percentage is 1.82%.
  • To convert to a decimal: Divide by 100.
  • Thus, 1.82% becomes 0.0182.
This rate is crucial for evaluating how fast a population rises over time, shaping predictions and planning for the future.
Exponential Growth Formula
The exponential growth formula is a powerful mathematical tool used to predict future values in situations of continuous growth.This formula is generally represented as:\[ N(t) = N_0 \times e^{rt} \]where:
  • - \(N(t)\): The future (predicted) population size
  • - \(N_0\): The initial population size
  • - \(e\): The base of natural logarithms, approximately 2.71828
  • - \(r\): The continuous growth rate expressed in decimals
  • - \(t\): The time period over which the growth is measured
By substituting the known values in the formula, it allows us to estimate future population sizes.In our example, we used India's initial population, continuous growth rate, and the number of years to predict the population size decades later.
India Population Estimate
Estimating the population of India using exponential growth involves projecting past data into the future. This exercise uses the given parameters to calculate the expected population: India's population was estimated at 766 million in 1985 with a continuous growth rate of 1.82% per year.

The steps to estimate the population in 2015 are:
  • Determine the initial population and growth rate.
  • Calculate the time interval from 1985 to 2015, which is 30 years.
  • Apply the exponential growth formula: \( N(t) = 766 \times e^{0.0182\times 30} \).
  • Compute the exponent and evaluate using a calculator to find the exponential term.
  • Finally, multiply to estimate the population, resulting in approximately 1322.516 million.
This method assumes the growth rate remains constant over the measured period, which is a simplification but useful for planning and analysis.

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Most popular questions from this chapter

Land value In 1867 the United States purchased Alaska from Russia for \(\$ 7,200,000\). There is 586,400 square miles of land in Alaska. Assuming that the value of the land increases continuously at \(3 \%\) per year and that land can be purchased at an equivalent price, determine the price of 1 acre in the year 2010 . (One square mile is equivalent to 640 acres.)

Exer. 35-38: Use common logarithms to solve for \(x\) in terms of \(y\). $$ y=\frac{10^{x}+10^{-x}}{2} $$

Refer to Exercises 67-68 in Section 3.3. If \(v_{0}\) is the wind speed at height \(h_{0}\) and if \(v_{1}\) is the wind speed at height \(h_{1}\), then the vertical wind shear can be described by the equation $$ \frac{v_{0}}{v_{1}}=\left(\frac{h_{0}}{h_{1}}\right)^{P} $$ where \(P\) is a constant. During a one-year period in Montreal, the maximum vertical wind shear occurred when the winds at the 200 -foot level were \(25 \mathrm{mi} / \mathrm{hr}\) while the winds at the 35 -foot level were \(6 \mathrm{mi} / \mathrm{hr}\). Find \(P\) for these conditions.

Exer. 47-50: Chemists use a number denoted by \(\mathrm{pH}\) to describe quantitatively the acidity or basicity of solutions. By definition, \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]\), where \(\left[\mathrm{H}^{+}\right]\)is the hydrogen ion concentration in moles per liter. Many solutions have a \(\mathrm{pH}\) between 1 and 14. Find the corresponding range of \(\left[\mathrm{H}^{+}\right]\).

The loudness of a sound, as experienced by the human ear, is based on its intensity level. A formula used for finding the intensity level \(\alpha\) (in decibels) that corresponds to a sound intensity \(I\) is \(\alpha=10 \log \left(I / I_{0}\right)\), where \(I_{0}\) is a special value of \(I\) agreed to be the weakest sound that can be detected by the ear under certain conditions. Find \(\alpha\) if (a) \(I\) is 10 times as great as \(I_{0}\) (b) \(I\) is 1000 times as great as \(I_{0}\) (c) \(I\) is 10,000 times as great as \(I_{0}\) (This is the intensity level of the average voice.)
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