Question

The population of the United States was \(3.9\) million in 1790 and 178 million in 1960 . If the rate of growth is assumed proportional to the number present, what estimate would you give for the population in \(2000 ?\) (Compare your answer with the actual 2000 population, which was 275 million.)

Short answer

Estimated 2000 population is 440 million; actual was 275 million.

Step-by-step solution

Understanding the Growth Model

We assume that the population growth is exponential due to its proportionality to the current population. This model can be described by the equation: \[ P(t) = P_0 e^{rt} \]where \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population, \(r\) is the growth rate, and \(t\) is the time in years after the initial year.

Identify Initial Parameters

First, identify \(P_0\) and \(P(t)\) from the data. For the year 1790, \(P_0 = 3.9\) million, and for the year 1960, \(P(170) = 178\) million (since it is 170 years after 1790). We need to estimate \(P(210)\), the population in 2000.

Calculate the Growth Rate

Using the equation for population growth:\[ 178 = 3.9 e^{170r} \]Solve for \(r\):\[ e^{170r} = \frac{178}{3.9} \]\[ 170r = \ln\left(\frac{178}{3.9}\right) \]\[ r = \frac{\ln\left(\frac{178}{3.9}\right)}{170} \]

Solving for Growth Rate

Calculate \(r\):\[ \frac{178}{3.9} \approx 45.64 \]\[ \ln(45.64) \approx 3.82 \]\[ r \approx \frac{3.82}{170} \approx 0.0225 \] or 2.25% per year.

Estimate the Population in 2000

Use the population growth formula to find \(P(210)\):\[ P(210) = 3.9 e^{0.0225 \times 210} \]Calculate:\[ e^{0.0225 \times 210} \approx e^{4.725} \approx 112.98 \]Thus, \[ P(210) = 3.9 \times 112.98 \approx 440\] million.

Compare with Actual Population

The calculated population in 2000 using this model is 440 million, considerably higher than the actual population of 275 million. This discrepancy suggests that the assumptions of constant growth rate were not met over the entire duration.

Key concepts

Population Modeling

Population modeling is a method used to estimate how the size of a population will change over time, using mathematical equations. This can be useful for predicting future population sizes and understanding potential impacts of changes in the population.

In this context, we are looking at a model where the population grows exponentially. This means the growth of the population at any time is proportional to its size at that time. To describe this mathematically, we use the function \[ P(t) = P_0 e^{rt}\] where

• \(P(t)\) is the population at time \(t\),
• \(P_0\) is the initial population size,
• \(r\) is the growth rate, and
• \(t\) is the time elapsed in years.

In our example, we start with 3.9 million people in 1790, and need to predict how the population changes by 2000.

Growth Rate Calculation

Calculating the growth rate is an essential step in modeling exponential growth, as it tells us how fast the population is increasing. When dealing with exponential growth, it is about finding the rate at which the population grows in reference to the current population.

Using the equation \[178 = 3.9 e^{170r},\]we solve for \(r\), the growth rate. Here's how:

• First, isolate the exponential by dividing both sides by the initial population: \[ e^{170r} = \frac{178}{3.9} \approx 45.64.\]
• Find the natural logarithm of both sides: \[ 170r = \ln(45.64) \approx 3.82.\]
• Finally, solve for \(r\) by dividing by 170:\[ r = \frac{3.82}{170} \approx 0.0225 \text{ or } 2.25\% \text{ per year}. \]

This step gives us a growth rate that can be used to forecast future populations.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, which makes them incredibly useful in solving problems involving exponential growth.

In the context of population growth, you might need to use logarithms to solve for the growth rate from an equation where population is expressed as an exponential function.

When you have an equation like \[178 = 3.9 e^{170r},\] you can solve for \(r\) by making the exponential term the subject of the equation and applying the natural logarithm:

• Convert the equation: \[ e^{170r} = \frac{178}{3.9}. \]
• Take the natural logarithm of both sides to get: \[ 170r = \ln\left(\frac{178}{3.9}\right). \]
• Finally, solve for \(r\): \[ r = \frac{\ln\left(\frac{178}{3.9}\right)}{170}. \]

This process illustrates how logarithmic functions help simplify solving exponential equations by transforming multiplication into addition, making complex exponential relationships much more manageable.