Problem 44 - Applied Calculus

Question

Which relative change is bigger in magnitude? Justify your answer. The change in the US population from 5.2 million to 7.2 million from 1800 to \(1810 ;\) the change in the US population from 151.3 to 179.3 from 1950 to \(1960 .\)

Step-by-Step Solution

Answer

The relative change from 1800 to 1810 is bigger, approximately 38.46%.

Step 1: Understanding Relative Change

Relative change is calculated as the increase or decrease in quantity from the initial value to the final value, divided by the initial value, and is often expressed as a percentage.

Step 2: Calculate Relative Change for 1800 to 1810

Calculate the relative change in the population from 1800 to 1810. The initial population is 5.2 million and the final population is 7.2 million. \[ \text{Relative Change (1800-1810)} = \frac{7.2 - 5.2}{5.2} \times 100\% \]Calculate the difference: 7.2 - 5.2 = 2.0. So,\[ \text{Relative Change (1800-1810)} = \frac{2.0}{5.2} \times 100\% \approx 38.46\% \]

Step 3: Calculate Relative Change for 1950 to 1960

Calculate the relative change in the population from 1950 to 1960. The initial population is 151.3 million and the final population is 179.3 million. \[ \text{Relative Change (1950-1960)} = \frac{179.3 - 151.3}{151.3} \times 100\% \]Calculate the difference: 179.3 - 151.3 = 28.0. So,\[ \text{Relative Change (1950-1960)} = \frac{28.0}{151.3} \times 100\% \approx 18.50\% \]

Step 4: Compare the Relative Changes

We have two relative changes calculated: - 1800 to 1810: approximately 38.46% - 1950 to 1960: approximately 18.50% Since 38.46% is greater than 18.50%, the relative change from 1800 to 1810 is bigger in magnitude.

Key Concepts

Understanding Percentage Change

Percentage change is a key concept used in understanding how the size of a quantity has changed with respect to its original value. This is crucial in many fields such as economics, business, and science. To calculate it, determine the difference between the final value and the initial value. This difference is then divided by the initial value. The result is multiplied by 100 to convert it into a percentage.

In mathematical terms, percentage change is represented as:
\[ \text{Percentage Change} = \frac{\text{Final Value} - \text{Initial Value}}{\text{Initial Value}} \times 100\% \]

• If the percentage change is positive, the quantity has increased.
• If the percentage change is negative, the quantity has decreased.

For example, when calculating the population growth for the US from 1800 to 1810, through this formula, the relative change was calculated to be approximately 38.46%, indicating a significant population increase. This calculation helps provide a clear and concise way to communicate the magnitude and direction of change over time.

Exploring Population Growth

Population growth refers to the increase in the number of individuals in a population. This natural process can be examined over different periods to understand demographic trends and patterns.

Changes in population numbers can stem from various factors, including births, deaths, and migration. Studying population growth helps in resource planning and policy formulation due to:

• Increased demand for amenities like food, housing, and healthcare.
• Implications on employment and economic development.
• Environmental impact due to larger population sizes.

Population growth can be expressed as a relative change, and it is often converted into a percentage change, as this allows for easy comparison between different periods or regions. For example, the US population's growth of 2.0 million people from 1800 to 1810, resulting in a 38.46% increase, highlights how rapidly the population increased during that decade compared to just an 18.50% increase from 1950 to 1960.

Importance of Historical Population Data

Historical population data plays a crucial role in understanding past demographic changes and informing future projections. By analyzing historical records, we can learn about the conditions and trends that shaped societies.

This data helps:

• In identifying historical patterns of population growth and decline.
• In providing insights into the effects of historical events such as wars, famines, and epidemics on populations.
• Planners and policymakers devise strategies to address future demographic shifts.

Historical population data often reveals influential periods of change, such as the early 1800s in the US. This era saw notable population growth due to factors such as westward expansion and immigration. The study of these patterns, like those between 1800 to 1810, compared to 1950 to 1960, helps highlight how historical context influences population changes. Understanding these trends enables better planning for infrastructure, resources, and services needed to support growing populations.