Problem 17 - Applied Calculus

Question

Table 1.16 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? $$\begin{array}{c|c|c|c} \hline \text { Year } & 1995 & 2000 & 2005 \\ \hline P \text { (billions) } & 5.68 & 6.07 & 6.45 \\ \hline A \text { (millions) } & 36.1 & 41.3 & 45.9 \\ \hline C \text { (millions) } & 91 & 740 & 2168 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

(a) Population: 0.077 billion/yr, Cars: 0.98 million/yr, Phones: 207.7 million/yr. (b) (i) Population increased faster; (ii) Cell phone subscribers increased faster.

Step 1: Understanding the Average Rate of Change

The average rate of change of a quantity over a time period is calculated as the change in the quantity divided by the change in time. We use the formula \( \text{Average Rate of Change} = \frac{\text{Final Value} - \text{Initial Value}}{\text{Time Period}} \). The units for the rate of change are derived from the quantity's units divided by time.

Step 2: Calculating for Population (P)

For world population \(P\):- Initial population (1995): 5.68 billion- Final population (2005): 6.45 billion- Time period: 10 yearsUsing the formula: \[ \text{Average Rate of Change} = \frac{6.45 - 5.68}{10} = \frac{0.77}{10} = 0.077 \text{ billion people/year} \]

Step 3: Calculating for Automobile Production (A)

For world automobile production \(A\):- Initial production (1995): 36.1 million cars- Final production (2005): 45.9 million cars- Time period: 10 yearsUsing the formula: \[ \text{Average Rate of Change} = \frac{45.9 - 36.1}{10} = \frac{9.8}{10} = 0.98 \text{ million cars/year} \]

Step 4: Calculating for Cell Phone Subscribers (C)

For world cell phone subscribers \(C\):- Initial subscribers (1995): 91 million- Final subscribers (2005): 2168 million- Time period: 10 yearsUsing the formula: \[ \text{Average Rate of Change} = \frac{2168 - 91}{10} = \frac{2077}{10} = 207.7 \text{ million subscribers/year} \]

Step 5: Comparing Population and Automobiles

Compare the average rate of change for population \(P\) and automobile production \(A\):- Population: 0.077 billion people/year- Automobiles: 0.98 million cars/year, or 0.00098 billion cars/yearSince 0.077 billion > 0.00098 billion, the population increased faster than the number of automobiles.

Step 6: Comparing Population and Cell Phone Subscribers

Compare the average rate of change for population \(P\) and cell phone subscribers \(C\):- Population: 0.077 billion people/year- Cell phones: 207.7 million subscribers/year, or 0.2077 billion subscribers/yearSince 0.2077 billion > 0.077 billion, the number of cell phone subscribers increased faster than the population.

Key Concepts

World Population

The world population has been steadily increasing over the years, and understanding the nuance behind its growth can be quite fascinating. The concept of average rate of change helps us measure how quickly population figures grow over a specific time frame. From the data provided, we can see that between 1995 and 2005, the global population grew from 5.68 billion to 6.45 billion.

Using the formula for average rate of change:\[\text{Average Rate of Change} = \frac{\text{Final Value} - \text{Initial Value}}{\text{Time Period}}\]This calculates the rate at which the population grew over 10 years. Calculating this, we get an increase of 0.077 billion people per year, or 77 million people per year. This increase in world population reflects not only natural growth but also shifts in global demographics.

Factors influencing population growth include:

• Improved healthcare systems leading to longer life expectancies.
• Variations in birth and death rates across different regions.
• Migration trends impacting population sizes of countries.

Understanding these changes is crucial for planning resources, such as food supply and healthcare infrastructure, to effectively accommodate the growing number of people.

Automobile Production

Automobile production is a key indicator of economic progress, aligning with industrial growth and technological advancements. The data shows that automobile production increased from 36.1 million cars in 1995 to 45.9 million cars by 2005.

By calculating the average rate of change:\[\text{Average Rate of Change} = \frac{45.9 \text{ million cars} - 36.1 \text{ million cars}}{10 \text{ years}} = 0.98 \text{ million cars/year}\]This tells us that annually, the production increased by 0.98 million cars, showcasing significant expansion in the automotive industry during that period.

Key drivers of this production trend include:

• Technological advancements leading to more efficient manufacturing processes.
• Increased consumer demand fueled by higher disposable incomes.
• Global trade expansion providing wider market access.

Understanding how these elements contribute to production change aids industries in planning, forecasting, and meeting future consumer demands effectively.

Cell Phone Subscribers

The increase in cell phone subscribers between 1995 and 2005 is a striking example of technological adoption and market expansion. In 1995, there were 91 million cell phone users worldwide, compared to a staggering 2,168 million (or 2.168 billion) by 2005.

This translates into an average rate of change calculated as:\[\text{Average Rate of Change} = \frac{2168 \text{ million subscribers} - 91 \text{ million subscribers}}{10 \text{ years}} = 207.7 \text{ million subscribers/year}\]This remarkable growth, adding approximately 207.7 million subscribers each year, highlights a period of rapid technological penetration.

Several factors contributed to the swift uptake of mobile technology:

• Innovations in telecommunications infrastructure making services more accessible and affordable.
• The increasing importance of mobile communication for personal and professional activities.
• Lower barriers to ownership such as cost reductions and diversified handset offerings.

The widespread adoption of cell phones exemplifies a significant shift in how people connect, work, and live, profoundly altering social and economic landscapes globally.