Question

Use \(y=y_{0} e^{k t}\). The population of Cairo grew from 5 million to 10 million in 20 years. Use an exponential model to find when the population was 8 million.

Short answer

The population was 8 million approximately 13.5 years after the start.

Step-by-step solution

Understand the Exponential Growth Model

We are given the exponential growth equation \( y = y_{0} e^{k t} \), where \( y \) is the population at time \( t \), \( y_{0} \) is the initial population, \( e \) is the base of natural logarithms, and \( k \) is the growth rate.

Set Up Known Values

Given that the population grew from 5 million to 10 million in 20 years, we know: \( y_{0} = 5 \) million, \( y = 10 \) million, and \( t = 20 \) years. Substitute these values into the formula: \( 10 = 5 e^{20k} \).

Solve for Growth Rate (k)

First, divide both sides by 5: \( 2 = e^{20k} \). Take the natural logarithm of both sides to eliminate the exponent: \( \ln(2) = 20k \). Solve for \( k \) by dividing both sides by 20: \( k = \frac{\ln(2)}{20} \).

Find Time when Population is 8 Million

We need to find \( t \) when the population \( y \) is 8 million. Use the equation \( 8 = 5 e^{kt} \). Substitute the value of \( k \) found previously to get: \( 8 = 5 e^{\left(\frac{\ln(2)}{20}\right)t} \).

Solve for Time (t)

Solve the equation for \( t \): Divide both sides by 5, \( \frac{8}{5} = e^{\left(\frac{\ln(2)}{20}\right)t} \). Take the natural logarithm of both sides: \( \ln\left(\frac{8}{5}\right) = \left(\frac{\ln(2)}{20}\right) t \). Solve for \( t \) by multiplying both sides by \( \frac{20}{\ln(2)} \): \( t = \frac{20 \ln\left(\frac{8}{5}\right)}{\ln(2)} \).

Calculate the Time (t)

Use a calculator to find \( t \): \( t \approx \frac{20 \times 0.470}\{0.693} \approx 13.5 \). Thus, it took approximately 13.5 years for the population to reach 8 million.

Key concepts

Population Growth

Population growth is an increase in the number of individuals in a region over time. When it comes to cities like Cairo, understanding how fast a population grows is crucial for planning and resource allocation. In mathematics, we often use models to estimate this growth over a specific period.

Exponential growth models are especially useful. They help predict future population based on current trends and past data. In these models, growth rate and time play key roles in determining the outcome, allowing us to make predictions about future population sizes.

In urban settings, factors like birth rates, death rates, and migration contribute to population growth. When using mathematical models, such changes over time are considered to provide accurate estimates.

Natural Logarithm

The natural logarithm is a fundamental concept in mathematics, represented by the symbol \( \ln \). It is the logarithm to the base \( e \), where \( e \approx 2.71828 \), an irrational and transcendental number.

In exponential growth models, natural logarithms are used to "undo" exponential functions. For example, if we have \( e^{x} = a \), taking the natural logarithm on both sides gives \( \ln(e^x) = \ln(a) \). Since \( \ln(e^x) = x \), it simplifies to \( x = \ln(a) \).

This application is crucial when solving for unknowns like the growth rate or time in population growth equations. Natural logarithms simplify the process and make it possible to translate exponential equations into a linear form that is more manageable.

Growth Rate Calculation

Growth rate calculation is a key aspect of analyzing population changes using exponential models.

When we know the initial population and the population after a given time, we can calculate the growth rate \( k \). This constant allows us to describe how quickly or slowly the population is increasing.

In our exercise example, the population doubled from 5 million to 10 million in 20 years. By using the formula:

• Start with the equation: \( 10 = 5 e^{20k} \)
• Divide both sides: \( 2 = e^{20k} \)
• Apply the natural logarithm: \( \ln(2) = 20k \)
• Solve for \( k \): \( k = \frac{\ln(2)}{20} \)

The calculated growth rate \( k \) is then used in further predictions and calculations. This rate helps understand how fast populations can grow under consistent conditions.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions model processes where a quantity grows at a rate proportional to its current value.

In population growth, the model \( y = y_0 e^{kt} \) is an exponential function. Here, \( y_0 \) is the initial population, \( e \) is the base of natural logarithms, \( k \) represents the growth rate, and \( t \) is time.

These functions are powerful in describing real-world phenomena. They allow us to model how populations can grow rapidly over time, given a consistent growth rate. Exponential functions are not only used in population studies but also in fields like finance, physics, and biology to describe growth and decay processes.