Problem 27 - Applied Calculus

Question

The infrastructure needs of a region (for example, the number of miles of electrical cable, the number of miles of roads, the number of gas stations) depend on its population. Cities enjoy economies of scale. \(^{90}\) For example, the number of gas stations is proportional to the population raised to the power of 0.77 (a) Write a formula for the number, \(N\), of gas stations in a city as a function of the population, \(P\), of the city. (b) If city \(A\) is 10 times bigger than city \(B\), how do their number of gas stations compare? (c) Which is expected to have more gas stations per person, a town of 10,000 people or a city of 500,000 people?

Step-by-Step Solution

Answer

1. (a) \(N = k \cdot P^{0.77}\); (b) City A has approximately 5.92 times more gas stations; (c) The town of 10,000 has more gas stations per person.

Step 1: Understanding the Relationship

The problem states that the number of gas stations, \(N\), is proportional to the population, \(P\), raised to the power of 0.77. This suggests a formula in the form \(N = k \cdot P^{0.77}\), where \(k\) is a proportionality constant. This is the mathematical representation of the given proportion.

Step 2: Relationship for Two Cities

Assume city \(A\) has a population \(P_A\) and city \(B\) has \(P_B = P_A/10\). According to the formula, the number of gas stations in city \(A\) is \(N_A = k \cdot P_A^{0.77}\), and for city \(B\) is \(N_B = k \cdot P_B^{0.77} = k \cdot \left( \frac{P_A}{10} \right)^{0.77}\). Simplifying, \(N_B = k \cdot \frac{P_A^{0.77}}{10^{0.77}}\). The ratio \(\frac{N_A}{N_B}\) shows how many times more gas stations city \(A\) has compared to city \(B\).

Step 3: Calculating the Ratio of Gas Stations

Calculate the ratio: \( \frac{N_A}{N_B} = \frac{k \cdot P_A^{0.77}}{k \cdot \left(\frac{P_A}{10}\right)^{0.77}} = 10^{0.77} \). Find \(10^{0.77}\) using a calculator to determine exactly how the number of gas stations compares.

Step 4: Comparing Per Capita Gas Stations

For a town with 10,000 people, \(N_{10000} = k \cdot 10000^{0.77}\). For a city with 500,000 people, \(N_{500000} = k \cdot 500000^{0.77}\). Calculate \(\frac{N_{10000}}{10000}\) and \(\frac{N_{500000}}{500000}\) to compare the number of gas stations per person for each location. This shows which has more gas stations per capita.

Key Concepts

Economies of Scale

Economies of scale is a concept within applied calculus and economics that describes how the average costs of production or services decrease as the size of the cities or firms increase. In the case of gas stations, this means that larger cities can have fewer gas stations per person because the infrastructure and service costs are spread over a larger population.
This occurs because as the population increases, the infrastructure required to serve each individual does not grow at the same rate, allowing for efficiency and cost savings.

• Larger populations can utilize shared resources more effectively.
• Costs can be spread over a greater number of individuals, reducing the per capita cost.

When applied to the problem at hand, we see that the number of gas stations is proportional to the population raised to a power of less than one, specifically 0.77, reflecting these economies of scale.
This reveals that larger cities do not need gas stations to increase proportionally with their population, allowing them to operate more efficiently.

Mathematical Modeling

Mathematical modeling in applied calculus involves creating equations to represent real-world situations. It allows us to predict and analyze various phenomena using mathematical concepts. In this exercise, we model the relationship between the population of a city and the number of gas stations available.
The equation given is: \( N = k \cdot P^{0.77} \) where \( N \) represents the number of gas stations and \( P \) is the population.

• \( k \) is a proportionality constant, encapsulating factors like urban planning and city structure.
• The exponent 0.77 shows how the number of gas stations grows in relation to population size, illustrating non-linear growth.

This model helps city planners understand infrastructure needs and optimize resources. By applying this equation, we can directly compare two cities' infrastructure strategies based on their size and predict how many gas stations a larger city will have compared to a smaller one.

Population Dynamics

Population dynamics is the study of how and why populations change over time and space. In applied calculus, it involves using mathematical methods to analyze these changes and their implications. In cities, population dynamics significantly affect infrastructure like gas stations.
When the population of a city grows, it doesn't merely require more infrastructure in a linear fashion. Instead, certain infrastructures like gas stations grow at a sublinear rate, a concept captured by the exponent 0.77 in our model.

• A city that is 10 times larger doesn't need 10 times the gas stations due to population dynamics favoring resource sharing.
• This affects per capita resource availability, leading to fewer gas stations per person in larger cities as compared to smaller ones.

Understanding these patterns helps in planning for sustainable growth and efficient service delivery, catering to dynamic changes in population while maintaining low operational costs.