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Chapter 4: Problem 11

These exercises use the population growth model. The population of the world was 5.7 billion in 1995 and the observed relative growth rate was 2% per year. (a) By what year will the population have doubled? (b) By what year will the population have tripled?

Short Answer

Expert verified
(a) By 2029 for doubling; (b) By 2050 for tripling.

Step by step solution

01

Understanding the Population Growth Formula

The population growth can be modeled using the exponential growth formula, \( P(t) = P_0 e^{rt} \), where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is time in years.
02

Inserting Known Variables for Doubling

Given \( P_0 = 5.7 \) billion and \( r = 0.02 \), we need to find the time \( t \) for the population to double. For doubling, \( P(t) = 2P_0 = 11.4 \) billion.
03

Setting up the Equation for Doubling

Substitute the values into the formula: \[ 11.4 = 5.7 e^{0.02t} \] Simplify to find \( e^{0.02t} \): \[ 2 = e^{0.02t} \]
04

Solving for Time Using Natural Logarithms (Doubling)

Take the natural logarithm on both sides to solve for \( t \): \[ \ln(2) = 0.02t \]Therefore, \[ t = \frac{\ln(2)}{0.02} \] Calculate \( t \).
05

Calculating Year for Doubling

Using \( \ln(2) \approx 0.693 \), we find \( t \approx \frac{0.693}{0.02} \approx 34.65 \) years. Adding this to 1995 gives the year approximately 2029.
06

Inserting Known Variables for Tripling

For tripling, \( P(t) = 3P_0 = 17.1 \) billion.
07

Setting up the Equation for Tripling

Substitute into the formula: \[ 17.1 = 5.7 e^{0.02t} \] Simplify to find \( e^{0.02t} \): \[ 3 = e^{0.02t} \]
08

Solving for Time Using Natural Logarithms (Tripling)

Take the natural logarithm on both sides to solve for \( t \): \[ \ln(3) = 0.02t \]Therefore, \[ t = \frac{\ln(3)}{0.02} \] Calculate \( t \).
09

Calculating Year for Tripling

Using \( \ln(3) \approx 1.0986 \), we find \( t \approx \frac{1.0986}{0.02} \approx 54.93 \) years. Adding this to 1995 gives the year approximately 2050.

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Key Concepts

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

Population Growth Model
The concept of population growth can be expressed through an exponential model. This model helps us understand how a population increases over time given a constant growth rate. Key components of this model:
  • **Initial Population** (\( P_0 \)): This is the starting number of individuals, here it's 5.7 billion in the year 1995.
  • **Growth Rate** (\( r \)): This is the constant rate at which the population increases, given as 2% or 0.02.
  • **Time Variable** (\( t \)): The variable that represents the time that has elapsed.
The exponential growth formula is \[ P(t) = P_0 e^{rt} \] Where:- \( P(t) \) is the population at time \( t \).- \( e \) is the base of the natural logarithm, approximately equal to 2.718.This model demonstrates how populations can grow rapidly over time if the growth rate remains constant. By rearranging the formula, you can solve for \( t \), helping to predict when the population will reach a certain size.
Natural Logarithms
Natural logarithms, denoted as \( \ln \), are the logarithms to the base 'e'.Why they're useful in exponential growth:
  • They help convert exponential equations into linear ones, making them easier to solve.
  • They are particularly useful for solving equations involving growth rates, such as the population growth formula.
For this exercise, natural logarithms are essential in solving for the time variable. When the population doubles, we use the equation:\[ \ln(2) = 0.02t \]By taking the natural logarithm of both sides, we can isolate \( t \). Similarly, when the population triples, the equation:\[ \ln(3) = 0.02t \]allows us to solve for \( t \) using natural logarithms. This simplifies finding the number of years it takes for the population to double or triple.
Growth Rate
The growth rate in population models is a crucial factor. It's the pace at which the population increases. For our model, the growth rate is 2%, which is expressed as 0.02 in the exponential growth formula.Understanding the growth rate:
  • It's usually expressed as a percentage.
  • In mathematical equations, it's written in decimal form.
  • It remains constant over the time period being considered.
In our example, a 2% annual growth suggests that every year, the population increases by 2% of the previous year's population.Calculating the specific time at which certain growth milestones are reached, like doubling or tripling the population, relies on knowing this growth rate. The exponential formula uses this rate: \[ P(t) = P_0 e^{rt} \]Where \( r \) is the growth rate. The predictions of reaching certain population sizes critically depend on this rate staying constant.

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Most popular questions from this chapter

Suppose you are offered a job that lasts one month, and you are to be very well paid. Which of the following methods of payment is more profitable for you? (a) One million dollars at the end of the month (b) Two cents on the first day of the month, 4 cents on the second day, 8 cents on the third day, and, in general, \(2^{n}\) cents on the \(n\) th day

A 15-g sample of radioactive iodine decays in such a way that the mass remaining after \(t\) days is given by \(m(t)=15 e^{-0.087 t}\) where \(m(t)\) is measured in grams. After how many days is there only 5 g remaining?

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These exercises use Newton鈥檚 Law of Cooling. A hot bowl of soup is served at a dinner party. It starts to cool according to Newton's Law of Cooling so that its temperature at time \(t\) is given by $$T(t)=65+145 e^{-0.05 t}$$ where \(t\) is measured in minutes and \(T\) is measured in \(^{\circ} \mathrm{F}\). (a) What is the initial temperature of the soup? (b) What is the temperature after 10 min? (c) After how long will the temperature be \(100^{\circ} \mathrm{F} ?\)

Find, correct to two decimal places, (a) the intervals on which the function is increasing or decreasing, and (b) the range of the function. $$y=10^{x-x^{2}}$$
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