Problem 26 - Applied Calculus

Question

Zipf's Law, developed by George Zipf in \(1949,\) states that in a given country, the population of a city is inversely proportional to the city's rank by size in the country. Assuming Zipf's Law: (a) Write a formula for the population, \(P\), of a city as a function of its rank, \(R\) (b) If the constant of proportionality \(k\) is 300,000 , what is the approximate population of the largest city (rank 1)? The second largest city (rank 2)? The third largest city? (c) Answer the questions of part (b) if \(k=6\) million. (d) Interpret the meaning of the constant of proportionality \(k\) in this context.

Step-by-Step Solution

Answer

(a) Formula: \(P = \frac{k}{R}\). (b) 300,000, 150,000, and 100,000. (c) 6,000,000, 3,000,000, and 2,000,000. (d) \(k\) is the population of the largest city.

Step 1: Formula for Population

According to Zipf's Law, the population of a city \(P\) is inversely proportional to its rank \(R\). This can be expressed mathematically as: \ P = \frac{k}{R} \ where \(k\) is the constant of proportionality.

Step 2: Population with k = 300,000

Using the formula \(P = \frac{k}{R}\), substitute \(k = 300,000\) to find the population for the largest cities: - Largest city (rank 1): \ P = \frac{300,000}{1} = 300,000 - Second largest city (rank 2): \ P = \frac{300,000}{2} = 150,000 - Third largest city (rank 3): \ P = \frac{300,000}{3} = 100,000.

Step 3: Population with k = 6,000,000

Substitute \(k = 6,000,000\) into the formula to find the population of the largest cities: - Largest city (rank 1): \ P = \frac{6,000,000}{1} = 6,000,000 - Second largest city (rank 2): \ P = \frac{6,000,000}{2} = 3,000,000 - Third largest city (rank 3): \ P = \frac{6,000,000}{3} = 2,000,000.

Step 4: Interpretation of k

The constant of proportionality \(k\) represents the population of the largest city when the rank is 1. It sets the scale for the population distribution, meaning larger \(k\) values result in higher populations overall for all ranks.

Key Concepts

Population Distribution

Population distribution is a concept used to describe the way in which population is spread over a specific area. It helps us understand the density and dispersion of people in cities and regions. According to Zipf's Law, the distribution of population across cities follows a particular pattern, where the size of a city's population is inversely proportional to its rank in terms of population size. This means that larger cities typically have significantly more people than smaller cities.
To see this in action, if we take the population distribution of cities in a country, we would observe that the largest city has the highest population, while each subsequent city's population decreases as its rank increases. This concept allows us to predict population sizes based on ranking, providing insights into how population density varies throughout regions.

Inverse Proportionality

Inverse proportionality is a relationship between two variables where an increase in one leads to a decrease in the other. In the context of Zipf's Law, the population of a city, denoted as \( P \), is inversely proportional to its rank \( R \), or \( P = \frac{k}{R} \). Here, \( k \) is a constant of proportionality.
This mathematical relationship indicates that as the rank \( R \) of a city increases, moving from a larger city to a smaller one, its population \( P \) decreases. It's fascinating to see how the application of this simple formula can provide meaningful approximations of population distributions in real-world scenarios.
Understanding this inverse relationship helps us grasp how population declines at a consistent rate as we move from the largest cities to smaller ones in a predictable pattern, outlined by the constant \( k \). This simplification is one of the beauties of mathematical modeling in describing real processes.

Ranking by Size

Ranking by size is crucial when applying Zipf's Law to understanding city populations. It involves ordering cities based on their populations from the largest to the smallest. This ranking process sets the stage for analyzing how populations change in relation to their rank. Using Zipf's formula, \( P = \frac{k}{R} \), where \( k \) denotes the constant of proportionality, we can illustrate how population sizes are determined.
Let's break down what this ranking tells us:

• The largest city, with rank 1, will always have a population equal to \( k \).
• The second largest city, rank 2, will have half the population of the largest city, as \( P = \frac{k}{2} \).
• Successive cities will have their populations decrease by factors defined by their ranks, revealing a predictable pattern of population declination.

This structured ranking by size aids not only in visualizing population shifts but also in anticipating urban population trends as cities grow and develop. It enhances our understanding of urban planning and resource allocation.