Problem 5 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Refer to Exercise \(4 .\) a. If the world population continues to grow at the rate of approximately \(2 \% / y e a r\), find the length of time \(t_{0}\) required for the world population to triple in size. b. Using the time \(t_{0}\) found in part (a), what would be the world population if the growth rate were reduced to \(1.8 \% / y e a r ?\)

Step-by-Step Solution

Answer

The time required for the world population to triple at a growth rate of 2% per year is approximately \(36.117\) years. If the growth rate is reduced to 1.8% per year for the same duration, the population would be about \(2.741\) times its initial size.

Step 1: Understand the exponential growth formula

The exponential growth formula is given by: \[P(t) = P_0(1 + r)^t\] where \(P(t)\) is the future population at time \(t\), \(P_0\) is the initial population, \(r\) is the growth rate per year, and \(t\) is the time in years.

Step 2: Find the time \(t_0\) for the world population to triple

According to the given problem, the rate of growth is \(2 \%\) per year. We need to find the time \(t_0\) when the population triples. Therefore, we can set up the equation as follows: \[3P_0 = P_0(1 + 0.02)^{t_0}\] Now, we need to solve for \(t_0\). First, divide both sides by \(P_0\): \[3 = (1 + 0.02)^{t_0}\] Next, take the natural logarithm (ln) of both sides: \(\ln(3) = \ln((1+ 0.02)^{t_0})\) Use the logarithm property \(\ln(a^b) = b\ln(a)\): \(\ln(3) = t_0\ln(1 + 0.02)\) Finally, solve for \(t_0\): \[t_0 = \frac{\ln(3)}{\ln(1 + 0.02)}\]

Step 3: Calculate the value of \(t_0\)

Using a calculator, compute the value of \(t_0\): \[t_0 = \frac{\ln(3)}{\ln(1.02)} \approx 36.117\] The time required for the world population to triple at a growth rate of \(2\%\) per year is approximately \(36.117\) years.

Step 4: Find the population if the growth rate is reduced to \(1.8\%\) per year

Now, we need to find the population that would result from reducing the growth rate to \(1.8 \%\) per year. To do this, use the same formula with the time \(t_0 \approx 36.117\) obtained in the previous step: \[P(t_0) = P_0(1 + 0.018)^{36.117}\] Since we are looking for the population ratio, we can simplify this expression by dividing both sides by \(P_0\): \[\frac{P(t_0)}{P_0} = (1 + 0.018)^{36.117}\]

Step 5: Calculate the population ratio with the reduced growth rate

Using a calculator, compute the population ratio: \[\frac{P(t_0)}{P_0} = (1 + 0.018)^{36.117} \approx 2.741\] The population ratio when the growth rate is reduced to \(1.8\%\) per year for \(36.117\) years is approximately 2.741. So, the world's population would be about \(2.741\) times its initial size if the growth rate were reduced to \(1.8 \%\) per year for the time it takes to triple at the \(2\%\) growth rate.

Key Concepts

Population Growth

Population growth refers to the change in the number of individuals in a population over time. It's an important concept in understanding how populations change and develop. Various factors contribute to this growth, including birth rates, death rates, immigration, and emigration.

When considering global population growth, two primary types are often discussed:

• Exponential Growth: Where the population grows at a constant percentage rate per year, leading to increasingly larger growth over time.
• Linear Growth: Where the population increases by a fixed number of individuals each year.

Exponential growth is more common in biological contexts due to the nature of reproduction. Understanding how this growth takes place helps governments and organizations plan for future needs and challenges, such as resource allocation and environmental conservation.

Exponential Growth Formula

The exponential growth formula is a mathematical expression used to describe how populations grow at a consistent rate over time. It is represented as: \[ P(t) = P_0(1 + r)^t \] Here's what each term means:

• \(P(t)\): The future population after time \(t\).
• \(P_0\): The initial population at the starting time.
• \(r\): The growth rate per year (expressed as a decimal).
• \(t\): The time in years for which the population has been growing.

This formula is crucial for predicting how long it will take for a population to reach a certain size, given a specific growth rate. For example, in the problem provided, the formula helps determine the time it would take for the world population to triple at different growth rates. It's a versatile tool applicable in various fields beyond demography, such as finance, biology, and environmental science.

Growth Rate Calculation

Calculating the growth rate involves determining how quickly a population is increasing or decreasing over a specific period. It's a key metric in many areas, from ecology to economics.

To calculate the growth rate, you can start by defining the rate as a percentage and then convert it to a decimal for use in mathematical models. For instance, a 2% growth rate would be represented as 0.02 in the exponential growth formula.

In practical terms, if you know the population triples at a 2% growth rate over approximately 36 years, you could also calculate what happens if the rate is adjusted. By using:

• \(r = 0.018\)
• \(t \approx 36.117\)

For this adjusted rate, the population's new ratio to the initial population can be calculated using: \[ \frac{P(t_0)}{P_0} = (1 + 0.018)^{36.117} \approx 2.741 \] This tells us the population would be about 2.741 times its initial size with the adjusted growth rate, providing valuable insights into population dynamics and planning.