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 world will reach its maximum population in approximately 75 years.

Step 1: Convert Arable Land to Acres

First, calculate the total arable land available in acres using the given figure of 13,500,000 square miles. Since there are 640 acres in a square mile, multiply: \[ 13,500,000 \text{ square miles} \times 640 \text{ acres/square mile} = 8,640,000,000 \text{ acres} \]

Step 2: Calculate Maximum Population Supported

Given that each person requires half an acre of land, we calculate the maximum population that can be supported. Using the available 8,640,000,000 acres, calculate: \[ \frac{8,640,000,000 \text{ acres}}{0.5 \text{ acres/person}} = 17,280,000,000 \text{ people} \] The land can thus support up to 17.28 billion people.

Step 3: Exponential Growth Equation

Use the exponential growth formula for population: \[ P(t) = P_0 \cdot e^{kt} \] where \(P_0\) is the initial population (6.4 billion), \(k\) is the growth constant (0.0132), and \(t\) is time in years. We want to find \(t\) when \(P(t) = 17.28\) billion people.

Step 4: Solve for Time

Rearrange and solve for \(t\): \[ e^{kt} = \frac{P(t)}{P_0} \] \[ e^{0.0132t} = \frac{17.28}{6.4} \approx 2.7 \] Take the natural logarithm on both sides: \[ 0.0132t = \ln(2.7) \approx 0.9933 \] \[ t = \frac{0.9933}{0.0132} \approx 75.3 \] years.

Key Concepts

Arable Land Conversion

To determine the maximum population that can be supported by the world's arable land, we first need to convert the land area into acres. Arable land refers to the land capable of being plowed and used to grow crops. It's crucial to know the amount of land available to produce food, especially when calculating how many people it can support.

In the problem, we are given

• 13,500,000 square miles of arable land.
• 1 square mile is equivalent to 640 acres.

Calculating the total acres is simple:First, multiply the number of square miles by the number of acres in each square mile:\[ 13,500,000 \times 640 = 8,640,000,000 \text{ acres} \]This conversion is crucial because it allows us to assess the agricultural potential of the land in terms of supporting population.

Exponential Growth Formula

The exponential growth formula is used to model situations where growth occurs at a consistent rate over time, such as population growth. The formula is:\[ P(t) = P_0 \cdot e^{kt} \]In this equation:

• \( P(t) \) is the population at time \( t \).
• \( P_0 \) is the initial size of the population, which is 6.4 billion in 2004.
• \( k \) is the growth constant, provided as 0.0132, indicating an annual growth rate.
• \( t \) is the time in years from the initial measurement.

This formula relies on the natural logarithm base \( e \), commonly used in continuous growth scenarios. By plugging in the respective values, we can solve for future population sizes or determine how long it will take to reach a specific population, as is required in our exercise.

Population Support Capacity

To determine how long the available land can support population growth, we must understand what population support capacity means. It's essentially the maximum number of people that can live in an area without running out of essential resources, like food. This problem assumes one person needs half an acre of land to be supported.Given:

• Total arable acres available: 8,640,000,000
• Half an acre needed per person

We find the maximum population by dividing the total acres by the land required per person:\[ \frac{8,640,000,000}{0.5} = 17,280,000,000 \text{ people} \]This calculation shows that the maximum sustainable population, given the available arable land and land need per person, is 17.28 billion.

Natural Logarithm Calculation

Once the exponential growth formula is set up, and we know the required future population, solving for time \( t \) involves using the natural logarithm. Here’s how that fits into solving our problem:After setting up the equation\[ e^{0.0132t} = \frac{17.28}{6.4} \approx 2.7 \]we need to isolate \( t \). Taking the natural logarithm on both sides allows for solving \( t \):\[ \ln(e^{0.0132t}) = \ln(2.7) \]This simplifies to:\[ 0.0132t = \ln(2.7) \]Given that \( \ln(2.7) \approx 0.9933 \), we can find \( t \) by dividing:\[ t = \frac{0.9933}{0.0132} \approx 75.3 \]This calculation shows that, considering the exponential growth rate, the global population would reach the maximum supportable level in approximately 75.3 years from 2004.