Problem 357 - Calculus Volume 1

Question

True or False? If true, prove it. If false, find the true answer. The populations of New York and Los Angeles are growing at 1\(\%\) and 1.4\(\%\) a year, respectively. Starting from 8 million (New York) and 6 million (Los Angeles), when are the populations equal?

Step-by-Step Solution

Answer

False. The populations will be equal in about 73 years.

Step 1: Understand the Problem

We need to determine when the populations of New York and Los Angeles will be equal based on given growth rates and initial populations.

Step 2: Set Up the Equations

The population of New York at time \( t \) is given by \( P_{NY}(t) = 8,000,000 \times (1 + 0.01)^t \). The population of Los Angeles is \( P_{LA}(t) = 6,000,000 \times (1 + 0.014)^t \). We need to find \( t \) such that \( P_{NY}(t) = P_{LA}(t) \).

Step 3: Equate the Population Formulas

Equate the expressions for the populations: \[ 8,000,000 \times (1.01)^t = 6,000,000 \times (1.014)^t \].

Step 4: Simplify the Equation

Divide both sides by \( 6,000,000 \) to simplify: \[ \frac{8}{6} \times (1.01)^t = (1.014)^t \]. This reduces to \[ \frac{4}{3} \times (1.01)^t = (1.014)^t \].

Step 5: Solve for \( t \) with Logarithms

Take the natural logarithm of both sides: \[ \ln\left(\frac{4}{3}\right) + t \cdot \ln(1.01) = t \cdot \ln(1.014) \]. Rearrange to find \( t \): \[ t = \frac{\ln\left(\frac{4}{3}\right)}{\ln(1.014) - \ln(1.01)} \].

Step 6: Calculate \( t \)

Calculate using a calculator: \( \ln\left(\frac{4}{3}\right) \approx 0.2877 \), \( \ln(1.014) \approx 0.0139 \), and \( \ln(1.01) \approx 0.00995 \). Substitute these values to get \[ t \approx \frac{0.2877}{0.0139 - 0.00995} \approx \frac{0.2877}{0.00395} \approx 72.82 \].

Step 7: Conclusion

It will take approximately 73 years for the populations to be equal, assuming no changes in growth rates.

Key Concepts

Population Growth

Population growth is a fascinating and crucial concept in the field of demography. It describes how the size of a population changes over time. In our case, we are examining the growth of the populations of New York and Los Angeles.
New York's population is currently growing at a rate of 1% per year. This means that every year, the population is increasing by 1% of its current size. Similarly, Los Angeles is growing at 1.4% annually.
Population growth can result from several factors:

• Natural increase (birth rate higher than death rate)
• Migration (people moving into and out of an area)

In our scenario, we are considering exponential growth. This means the growth sticks to a percentage of the current population size, resulting in the population increasing by larger numbers over time. This is important because exponential growth can lead to rapid increases in population over short periods.

Mathematical Modeling

Mathematical modeling allows us to represent real-world phenomena using mathematical formulas and concepts. In our exercise, each city's population growth is modeled using an exponential growth function.
This involves setting up equations that reflect how populations change:

• For New York: \( P_{NY}(t) = 8,000,000 \times (1 + 0.01)^t \)
• For Los Angeles: \( P_{LA}(t) = 6,000,000 \times (1 + 0.014)^t \)

Here, \( t \) represents time in years, \(8,000,000\) and \(6,000,000\) are the initial populations, and \(1.01\) and \(1.014\) are the growth factors.
One of the core strengths of mathematical modeling in this context is the ability to predict future population sizes based on current trends, which is essential for planning resources, infrastructure, and services.

Logarithms

Logarithms are powerful tools in mathematics, especially useful for solving equations involving exponentials. They are the inverse operations of exponentiation, just like division is the inverse of multiplication.
In our problem, logarithms help us isolate \( t \) when the population functions for New York and Los Angeles are equated:

• We start with \( 8,000,000 \times (1.01)^t = 6,000,000 \times (1.014)^t \).
• Simplify to \( \frac{4}{3} \times (1.01)^t = (1.014)^t \).

Taking the natural logarithm on both sides helps in simplifying the equation:

• \( \ln\left(\frac{4}{3}\right) + t \cdot \ln(1.01) = t \cdot \ln(1.014) \)

By rearranging, we solve for \( t \), giving us insight into when both populations will be equal. This demonstrates how logarithms can convert multiplicative processes into easier additive processes, making complex problems more manageable.