Problem 43 - Calculus Early Transcendentals: Pearson New International Edition

Question

Assume that (1) world population continues to grow exponentially with growth constant \(k=0.0132,(2)\) it takes \(\frac{1}{2}\) acre of land to supply food for one person, and (3) there are \(13,500,000\) square miles of arable land in the world. How long will it be before the world reaches the maximum population? Note: There were \(6.4\) billion people in 2004 and 1 square mile is 640 acres.

Step-by-Step Solution

Answer

The maximum population will be reached in approximately 75 years.

Step 1: Convert Arable Land to Acres

First, we need to convert the total arable land from square miles to acres. Since 1 square mile is equal to 640 acres, we calculate the total acres: \(13,500,000 \, \text{square miles} \times 640 \, \text{acres/square mile} = 8,640,000,000 \, \text{acres}\).

Step 2: Determine Maximum Population Supported by Land

According to the problem, 0.5 acres feeds one person. So, we divide the total number of acres by 0.5 to determine the maximum population that can be supported: \(\frac{8,640,000,000 \, \text{acres}}{0.5 \, \text{acres/person}} = 17,280,000,000 \, \text{people}\).

Step 3: Use the Exponential Growth Formula

The population grows exponentially according to the formula \(P(t) = P_0 e^{kt}\), where \(P_0\) is the initial population, \(k\) is the growth constant, and \(t\) is the time in years. Initially, \(P_0 = 6.4 \, \text{billion}\), and we want to find \(t\) when \(P(t) = 17.28 \, \text{billion}\).

Step 4: Substitute Values and Solve for Time

Substitute the known values into the formula: \(17.28 = 6.4e^{0.0132t}\). To solve for \(t\), first divide both sides by 6.4: \(e^{0.0132t} = \frac{17.28}{6.4}\). Then take the natural logarithm (ln) of both sides: \(0.0132t = \ln\left(\frac{17.28}{6.4}\right)\). Finally, solve for \(t\) by dividing both sides by 0.0132: \[t = \frac{\ln\left(\frac{17.28}{6.4}\right)}{0.0132}\].

Step 5: Calculate the Time

Calculate the division: \(\frac{17.28}{6.4} = 2.7\). Find the natural logarithm: \(\ln(2.7) \approx 0.9933\). Divide by the growth constant: \[t = \frac{0.9933}{0.0132} \approx 75.27\, \text{years}\].

Key Concepts

Arable Land

Arable land refers to land that can be used for agricultural purposes, which is vital for producing food to support human populations. Understanding the conversion of land measurements is crucial when working with large-scale land areas, as demonstrated in problems concerning agricultural capacity.

In the given exercise, we had to convert arable land from square miles to acres to determine how much land is available for agriculture.

• One square mile is equivalent to 640 acres.
• Thus, when estimating available agricultural land, it is important to use consistent units across calculations.
• This understanding allows us to calculate how many people the land can potentially support.

This premise of arable land uses areas expressed in acres, facilitating easy comparisons between land areas and the number of people they can sustain.

Population Dynamics

Population dynamics is the study of how and why populations change over time. The exercise concerned exponential growth, a common concept in population dynamics where the population grows continuously at a rate proportional to its current size.

Key aspects of exponential growth in population dynamics include:

• Initial Population: This is the starting number of people (6.4 billion in 2004).
• Growth Rate: Represented by the constant \(k\), this defines how quickly the population grows (\(k = 0.0132)\).
• Carrying Capacity: This refers to the maximum population the environment can sustain, based in this case on arable land availability.

Understanding these factors helps predict future population sizes and how soon they may exceed the land's carrying capacity.

Natural Logarithms

Natural logarithms (ln) are an important mathematical tool used in solving exponential growth equations. They help find the time required for a population to reach a certain size when growing exponentially.

In our exercise, we used natural logarithms to unravel the exponential equation:

• Given a growth equation \(P(t) = P_0 e^{kt}\), solving for \(t\) involved taking the natural log of both sides.
• This step simplifies the expression, making it easier to isolate and solve for \(t\) (e.g., \(\ln\left(\frac{17.28}{6.4}\right)\)).
• Natural logs convert exponential functions into manageable linear forms, essential for clear, accessible calculations.

Grasping this concept allows more efficient problem-solving in scenarios involving exponential changes, like population growth.