Problem 17 - Calculus Early Transcendentals

Question

According to the 2010 census, the U.S. population was 309 million with an estimated growth rate of \(0.8 \% / \mathrm{yr}\). a. Based on these figures, find the doubling time and project the population in 2050 . b. Suppose the actual growth rate is just 0.2 percentage point lower than \(0.8 \% / \mathrm{yr}(0.6 \%) .\) What are the resulting doubling time and projected 2050 population? Repeat these calculations assuming the growth rate is 0.2 percentage point higher than \(0.8 \% / \mathrm{yr}\). c. Comment on the sensitivity of these projections to the growth rate.

Step-by-Step Solution

Answer

Short Answer: At a growth rate of 0.8%, the doubling time is approximately 87.15 years, and the projected population in 2050 is 431.25 million. For a growth rate of 0.6%, the doubling time increases to around 115.53 years, and the projected population decreases to 400.47 million. On the other hand, for a growth rate of 1.0%, the doubling time decreases to about 69.66 years, and the projected population increases to 461.37 million. These changes in growth rate by 0.2% result in significant differences in doubling time and projected population, indicating that population projections and doubling time are sensitive to changes in the growth rate assumption.

Step 1: Doubling Time Formula

The formula to find the doubling time is given by: Doubling time (t) = \(\frac{ln(2)}{ln(1 + r)}\) where 'r' is the growth rate (as a decimal).

Step 2: Population Growth Formula

The formula to project population in the future is given by: Future Population (P) = Initial Population (P₀) * \((1 + r)^{t}\) P₀ = Initial Population t = years passed r = growth rate (as a decimal)

Step 3: PART a: Doubling Time & Population Projection for 0.8% Growth Rate

Given data: Initial Population (P₀) = 309 million Growth Rate (r) = 0.8% = 0.008 Step 1: Calculate Doubling Time (t) Doubling time (t) = \(\frac{ln(2)}{ln(1 + 0.008)}\) ≈ 87.15 years Step 2: Project Population in 2050 Years passed (t) = 2050 - 2010 = 40 years Future Population (P) = 309 * \((1 + 0.008)^{40}\) = 309 * 1.3953 = 431.25 million

Step 4: PART b: Doubling Time & Population Projections for 0.6% and 1.0% Growth Rates

Case 1: Growth rate (r) = 0.6% = 0.006 Doubling time (t) = \(\frac{ln(2)}{ln(1 + 0.006)}\) ≈ 115.53 years Future Population (P) = 309 * \((1 + 0.006)^{40}\) = 309 * 1.2955 = 400.47 million Case 2: Growth rate (r) = 1.0% = 0.010 Doubling time (t) = \(\frac{ln(2)}{ln(1 + 0.010)}\) ≈ 69.66 years Future Population (P) = 309 * \((1 + 0.010)^{40}\) = 309 * 1.4918 = 461.37 million

Step 5: PART c: Comment on Sensitivity of Projections to Growth Rate

As the growth rate changes by 0.2%, the doubling time and projected population also change significantly. For a decrease of 0.2% in growth rate, doubling time increases by around 28.38 years, and the projected population decreases by 30.78 million people. Conversely, for an increase of 0.2% in the growth rate, doubling time decreases by about 17.49 years, and the projected population increases by 30.12 million people. This shows that population projections and doubling time are sensitive to changes in the growth rate assumption.

Key Concepts

Doubling Time

In exponential growth, the concept of doubling time can be quite intriguing. This is the period it takes for a quantity to double in size or value. Often, it's associated with populations or investments. In mathematical terms, doubling time is determined by the formula:\[\text{Doubling time} = \frac{\ln(2)}{\ln(1 + r)}\]Here, \(r\) represents the growth rate expressed as a decimal. This formula reflects how quickly population or other metrics can escalate over time. For instance, at a growth rate of 0.8%, the U.S. population doubling time is approximately 87.15 years. Moreover, subtle alterations in the growth rate immensely impact doubling time. Lowering the rate to 0.6% lengthens the doubling time to approximately 115.53 years. Conversely, a 1.0% growth rate shortens it to about 69.66 years. Understanding these dynamics is crucial, especially for making accurate long-term projections.

Population Projection

Population projection is a mathematical estimation of future population sizes based on current data and assumed growth rates. Using the formula:\[P = P_0 \times (1 + r)^t\]where \(P_0\) is the initial population, \(r\) is the growth rate, and \(t\) is the number of years into the future we wish to project, one can estimate future population sizes.Taking the 2010 U.S. data as an example:

• Initial population \(P_0\) was 309 million.
• Estimated growth rate \(r\) was 0.8% or 0.008.
• For projections up to 2050, \(t\) is 40 years.

When these values are substituted, the projected population in 2050 is approximately 431.25 million. This projection shifts notably with variations in \(r\). If \(r\) is reduced to 0.6%, the 2050 projection falls to around 400.47 million. If it rises to 1.0%, it elevates to approximately 461.37 million. Such calculations highlight the significance of accurate data for policy planning and resource allocation.

Growth Rate Sensitivity

The sensitivity of growth projections to changes in the growth rate is a pivotal aspect of population demographics. Small alterations in growth rates substantially influence doubling times and future population projections. Consider the U.S. population scenario:

• A mere 0.2% decrease in \(r\) from 0.8% to 0.6% extends the doubling time by almost 28.38 years.
• The estimated population in 2050 decreases by about 30.78 million people.
• An increase of 0.2% to 1.0% reduces the doubling time by around 17.49 years, increasing the projected population by roughly 30.12 million people.

This sensitivity underscores the importance of precise growth rate measurements in demographic studies. Ignoring even slight inaccuracies can lead to significant divergences in population estimates, impacting urban planning, economic strategies, and environmental assessments.