Problem 29 - Applied Calculus

Question

The population of the world reached 1 billion in 1804 2 billion in 1927,3 billion in 1960,4 billion in 1974,5 billion in 1987 and 6 billion in \(1999 .\) Find the average rate of change of the population of the world, in people per minute, during each of these intervals (that is, from \(1804 \text { to } 1927,1927 \text { to } 1960, \text { etc. })\)

Step-by-Step Solution

Answer

Average rates: 15.44, 57.54, 135.76, 146.23, and 158.45 people/min.

Step 1: Identify the intervals and populations

Identify the population at the start and end of each interval, and note the years: - From 1804 to 1927, population changes from 1 billion to 2 billion. - From 1927 to 1960, population changes from 2 billion to 3 billion. - From 1960 to 1974, population changes from 3 billion to 4 billion. - From 1974 to 1987, population changes from 4 billion to 5 billion. - From 1987 to 1999, population changes from 5 billion to 6 billion.

Step 2: Calculate the time in minutes for each interval

Convert each interval into minutes:- 1804 to 1927: 123 years \(123 \times 365.25 \times 24 \times 60 = 64,781,160 \text{ minutes}\)- 1927 to 1960: 33 years \(33 \times 365.25 \times 24 \times 60 = 17,383,800 \text{ minutes}\)- 1960 to 1974: 14 years \(14 \times 365.25 \times 24 \times 60 = 7,365,840 \text{ minutes}\)- 1974 to 1987: 13 years \(13 \times 365.25 \times 24 \times 60 = 6,837,720 \text{ minutes}\)- 1987 to 1999: 12 years \(12 \times 365.25 \times 24 \times 60 = 6,307,920 \text{ minutes}\)

Step 3: Calculate the average rate of change for each interval

For each interval, use the formula \(\text{Rate} = \frac{\text{Change in population}}{\text{Change in time}} \):- 1804 to 1927: \(\frac{1,000,000,000}{64,781,160} \approx 15.44 \text{ people per minute}\)- 1927 to 1960: \(\frac{1,000,000,000}{17,383,800} \approx 57.54 \text{ people per minute}\)- 1960 to 1974: \(\frac{1,000,000,000}{7,365,840} \approx 135.76 \text{ people per minute}\)- 1974 to 1987: \(\frac{1,000,000,000}{6,837,720} \approx 146.23 \text{ people per minute}\)- 1987 to 1999: \(\frac{1,000,000,000}{6,307,920} \approx 158.45 \text{ people per minute}\)

Key Concepts

Population Growth

Population growth refers to the increase in the number of individuals in a population, which in this context is the world population over time. This growth is often measured over specific periods to understand trends, challenges, and opportunities.
When evaluating population growth from historical data, it is essential to look at:

• The size of the population at different points in time
• Factors contributing to these shifts, such as birth rates, mortality rates, and migration

The numbers provided in the exercise indicate a steady, though increasing rate of growth over nearly two centuries. With every passing interval, from 1804 to 1999, the number of people added within a specific timeframe increased, reflecting accelerated growth likely influenced by industrial advancements, improvements in healthcare, and agricultural productivity. Understanding population growth is crucial because it helps nations plan for resources, infrastructure, and services to cater to an increasing number of individuals.

Time Intervals

Time intervals are vital for analyzing how changes occur over specific periods. In the context of population growth, these intervals help us break down the timeline into manageable segments, enabling us to compare the changes accurately.
For this exercise, each interval represents a period between two significant milestones of population growth:

• 1804 to 1927
• 1927 to 1960
• 1960 to 1974
• 1974 to 1987
• 1987 to 1999

Calculating each time interval in minutes requires converting years into minutes. This detailed conversion involves accounting for leap years with a slight fraction over 365 days per year (365.25 days), then multiplying by 24 hours and 60 minutes. These conversions result in:

• From 1804 to 1927, we have 64,781,160 minutes.
• From 1927 to 1960, we have 17,383,800 minutes.
• From 1960 to 1974, we have 7,365,840 minutes.
• From 1974 to 1987, we have 6,837,720 minutes.
• From 1987 to 1999, we have 6,307,920 minutes.

Accurately measuring these intervals helps in the subsequent calculation of average rates of change, revealing vital insights into how swiftly populations are increasing over different times.

People per Minute

"People per minute" is a fascinating unit to calculate average rate of change, highlighting the pace of population growth over the specified time intervals. This calculation is performed by dividing the change in population by the change in time, measured in minutes.
This method provides a clear view of how quickly populations grew during distinct historical periods:

• Between 1804 to 1927, the rate was approximately 15.44 people per minute.
• From 1927 to 1960, it increased to around 57.54 people per minute.
• For 1960 to 1974, the rate rose further to about 135.76 people per minute.
• From 1974 to 1987, it became roughly 146.23 people per minute.
• Finally, from 1987 to 1999, the rate peaked at approximately 158.45 people per minute.

The increasing people per minute rate reflects how technological, medical, and social advancements contributed to sustaining larger populations. It also underscores the importance of monitoring these changes to address challenges such as resource allocation and environmental impact effectively. This measure is not only a statistical representation but a tool for planning and foresight into the future of global populations.