Problem 20 - Applied Calculus

Question

Table \(1.15\) shows world population, \(P\), in billions of people, world passenger automobile production, \(A\), in millions of cars, and world cell phone subscribers, \(C\), in millions of subscribers. \({ }^{33}\) (a) Find the average rate of change, with units, for each of \(P, A\), and \(C\) between 1995 and 2005 . (b) Between 1995 and 2005, which increased faster: (i) Population or the number of automobiles? (ii) Population or the number of cell phone subscribers? $$ \begin{array}{l} \text { Table } 1.15\\\ \begin{array}{c|c|c|c} \hline \text { Year } & 1995 & 2000 & 2005 \\ \hline P \text { (billions) } & 5.68 & 6.07 & 6.45 \\ \hline A \text { (millions) } & 36.1 & 41.3 & 45.9 \\ \hline C \text { (millions) } & 91 & 740 & 2168 \\ \hline \end{array} \end{array} $$

Step-by-Step Solution

Answer

Cars increased faster than population, and cell phone subscribers increased faster than population.

Step 1: Calculate Average Rate of Change for Population (P)

To find the average rate of change of the population, use the formula \( \frac{\Delta P}{\Delta t} = \frac{P_{\text{final}} - P_{\text{initial}}}{t_{\text{final}} - t_{\text{initial}}} \). For population, \( P_{\text{initial}} = 5.68 \) billion in 1995 and \( P_{\text{final}} = 6.45 \) billion in 2005. The time span is 2005 - 1995 = 10 years. Thus, the average rate of change is \( \frac{6.45 - 5.68}{10} = 0.077 \) billion people per year or 77 million people per year.

Step 2: Calculate Average Rate of Change for Automobiles (A)

Similar to step 1, calculate the average rate of change for world automobile production using \( A_{\text{initial}} = 36.1 \) million cars in 1995 and \( A_{\text{final}} = 45.9 \) million cars in 2005. The average rate of change is \( \frac{45.9 - 36.1}{10} = 0.98 \) million cars per year.

Step 3: Calculate Average Rate of Change for Cell Phone Subscribers (C)

For cell phone subscribers, start with \( C_{\text{initial}} = 91 \) million in 1995 and \( C_{\text{final}} = 2168 \) million in 2005. The average rate of change is \( \frac{2168 - 91}{10} = 207.7 \) million subscribers per year.

Step 4: Compare Rate of Increase Between Population and Automobiles

The rate of increase for the population is 77 million per year, while for automobiles it is 0.98 million per year. Therefore, compared in millions, the number of automobiles increased faster (0.98 > 0.077, since in direct millions it is more relevant than just billions due to the context).

Step 5: Compare Rate of Increase Between Population and Cell Phone Subscribers

Compare 77 million people per year with 207.7 million subscribers per year. The number of cell phone subscribers increased faster than the population (207.7 > 77).

Key Concepts

Understanding World Population Growth

World population growth is an important phenomenon that has significant implications for global resources and sustainability. To understand how quickly the world population is growing, we can calculate the average rate of change over a specified period.

From 1995 to 2005, the global population increased from 5.68 billion to 6.45 billion people. To find how much the population grew on average each year, we use the formula for average rate of change: \( \frac{\Delta P}{\Delta t} = \frac{P_{\text{final}} - P_{\text{initial}}}{t_{\text{final}} - t_{\text{initial}}} \).

Plugging in the values:

• Initial population in 1995: 5.68 billion
• Final population in 2005: 6.45 billion
• Time span: 10 years

The average increase per year is \( \frac{6.45 - 5.68}{10} = 0.077 \) billion people, which is equivalent to 77 million people each year.

Understanding this growth helps policymakers and scientists address challenges like resource allocation, urban planning, and environmental impact as the population continues to increase.

Changes in Automobile Production Over Time

Automobile production is another key indicator of technological progress and economic development worldwide. It reflects larger trends in industry and transportation. By calculating the average rate of change, we can see how production has evolved.

Between 1995 and 2005, automobile production increased from 36.1 million cars to 45.9 million cars globally. For the average rate of change, use the same mathematical approach:

\( \frac{A_{\text{final}} - A_{\text{initial}}}{t_{\text{final}} - t_{\text{initial}}} \)

Here's what the numbers look like:

• Initial production in 1995: 36.1 million cars
• Final production in 2005: 45.9 million cars
• Duration: 10 years

Thus, the average annual growth in production is \( \frac{45.9 - 36.1}{10} = 0.98 \) million cars per year.

This growth occurred at a faster pace compared to the population, indicating robust advancement in automotive technology and increased demand for vehicles during that period.

Surge in Cell Phone Subscribers

The explosive rise in cell phone subscribers from 1995 to 2005 highlights a key technological revolution. Mobile phones transitioned from luxury items to essential personal devices during this time.

The number of cell phone subscribers worldwide shot up from 91 million to a staggering 2,168 million. To assess the pace of change, apply the formula:

\( \frac{C_{\text{final}} - C_{\text{initial}}}{t_{\text{final}} - t_{\text{initial}}} \)

Consider these values:

• Subscribers in 1995: 91 million
• Subscribers in 2005: 2,168 million
• Time interval: 10 years

The average increase is \( \frac{2168 - 91}{10} = 207.7 \) million new subscribers per year.

This remarkable growth outpaced both the population increase and the growth in automobile production, underlining the critical role of digital connectivity in modern society.