Problem 17 - Calculus Early Transcendentals

Question

Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point\((t=0)\) and units of time. According to the 2010 census, the U.S. population was 309 million with an estimated growth rate of \(0.8 \%\) / yr. a. Based on these figures, find the doubling time and project the population in 2050. b. Suppose the actual growth rates are just 0.2 percentage points lower and higher than \(0.8 \% / \mathrm{yr}(0.6 \% \text { and } 1.0 \%) .\) What are the resulting doubling times and projected 2050 populations? c. Comment on the sensitivity of these projections to the growth rate.

Step-by-Step Solution

Answer

Answer: The projections are sensitive to the growth rate. A small change in the growth rate (±0.2 percentage points) causes a noticeable difference in doubling times (approximately 26-46 years) and a difference of 12-25 million people in the 2050 population projections.

Step 1: Convert Growth Rate to Decimal

Convert the given growth rate of 0.8% to decimal form: \(r = 0.008.\)

Step 2: Exponential Growth Function

Use the exponential growth formula with the given values: \(P(t)=309(1+0.008)^t.\)

Step 3: Find Doubling Time

Set \(P(t) = 2 \cdot 309\) and solve for \(t\): $$2 \cdot 309 = 309(1+0.008)^t.$$ $$t = \frac{\log(2)}{\log(1.008)} \approx 87.67 \text{ years}.$$

Step 4: Project the Population in 2050

Substitute \(t = 2050 - 2010 = 40\) in the population formula: $$P(40) = 309(1.008)^{40} \approx 385.55 \text{ million}.$$

Step 5: Adjust Growth Rate and Recalculate

Alter the growth rate by ±0.2 percentage points (0.6% and 1.0%) and convert them to decimals: \(r_1 = 0.006\) and \(r_2 = 0.01.\) Using these new growth rates, find their doubling times and project the 2050 population: For \(r_1 = 0.006\): - Doubling time: \(t = \frac{\log(2)}{\log(1.006)} \approx 115.89 \text{ years}\) - 2050 population: \(P(40)= 309(1.006)^{40} \approx 374.78 \text{ million}\) For \(r_2 = 0.01\): - Doubling time: \(t = \frac{\log(2)}{\log(1.01)} \approx 69.66 \text{ years}\) - 2050 population: \(P(40)= 309(1.01)^{40} \approx 399.97 \text{ million}\)

Step 6: Comment on Sensitivity of Projections

The projections are sensitive to the growth rate. A small change in the growth rate (±0.2 percentage points) causes a noticeable difference in doubling times (approximately 26-46 years) and a difference of 12-25 million people in the 2050 population projections. Real-world population growth rates can vary, so it is essential to consider potential variations in growth rates when making long-term predictions.

Key Concepts

Doubling Time

Doubling time is a fascinating concept in exponential growth that answers the question: "How long will it take for a quantity to double in size?" It is particularly useful in understanding population growth over time.
To calculate doubling time, we use the formula:
\[ t = \frac{\log(2)}{\log(1+r)} \]Where:

• \( t \) is the doubling time in years,
• \( r \) is the growth rate expressed as a decimal.

For example, with a growth rate of 0.8% or 0.008 per year, doubling time becomes about 87.67 years.
This means it will take roughly 88 years for the population to double from its starting point. Doubling time provides an intuitive grasp of the long-term effects of continuous growth, making it a vital tool for policy makers and planners.

Population Projection

Population projection is a method used to estimate future population size based on current data and assumptions about future growth.
We can use exponential growth formulas to make these projections. Using the formula:
\[ P(t) = P_0 (1 + r)^t \]Where:

• \( P(t) \) is the future population size,
• \( P_0 \) is the initial population size,
• \( r \) is the growth rate in decimal form,
• \( t \) is the time in years from the initial point.

For example, with an initial U.S. population of 309 million in 2010 and a growth rate of 0.8%, projecting to the year 2050 results in an estimated population of approximately 385.55 million.
Population projections help governments and organizations plan for future needs, such as infrastructure, health care, and education.

Growth Rate Sensitivity

Growth rate sensitivity refers to how sensitive an outcome is to changes in the growth rate.
Even small changes in growth rates can lead to large differences in long-term projections.
Consider the U.S. population with growth rates of 0.6% and 1.0%, compared to the original 0.8%. At 0.6%, the doubling time stretches to about 115.89 years and the 2050 projection drops to about 374.78 million people. Conversely, with a 1.0% growth rate, the doubling time shortens to approximately 69.66 years, and the 2050 projection increases to about 399.97 million.
This sensitivity highlights the importance of accurately estimating growth rates and considering a range of scenarios when making long-term forecasts.

• A slight increase or decrease in growth rate can significantly impact population estimates.
• Accurate data and realistic assumptions are crucial.

Understanding growth rate sensitivity is essential for reliable population projections and planning.