Problem 356 - Calculus Volume 1

Question

For the following exercises, 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.

Step-by-Step Solution

Answer

The population was 8 million approximately 13.5 years after the initial measurement.

Step 1: Define the exponential model

The formula for exponential growth is given by \( y = y_0 e^{k t} \), where \( y \) is the final amount, \( y_0 \) is the initial amount, \( k \) is the growth rate, and \( t \) is the time in years.

Step 2: Use given data to set up equations

From the data, we know \( y_0 = 5 \) million, \( y = 10 \) million, and \( t = 20 \) years. Substitute these values into the equation: \( 10 = 5 e^{20k} \).

Step 3: Solve for the growth rate \( k \)

Rearrange the equation to solve for \( k \):\[ \frac{10}{5} = e^{20k} \implies 2 = e^{20k} \]Taking the natural logarithm on both sides, we get:\[ \ln(2) = 20k \implies k = \frac{\ln(2)}{20} \].

Step 4: Use the growth rate to find time when the population was 8 million

Now, use the growth rate to find when the population was 8 million by substituting \( y = 8 \), \( y_0 = 5 \), and the calculated \( k \) into the model:\[ 8 = 5 e^{kt} \].

Step 5: Solve for time \( t \)

Rearrange the equation to solve for \( t \):\[ \frac{8}{5} = e^{kt} \implies e^{kt} = 1.6 \].Take the natural logarithm of both sides:\[ \ln(1.6) = kt \implies t = \frac{\ln(1.6)}{k} \].

Step 6: Calculate time \( t \)

Use the previously calculated value of \( k \):\[ k = \frac{\ln(2)}{20} \].Substitute in the formula: \[ t = \frac{\ln(1.6)}{\ln(2)/20} = 20 \cdot \frac{\ln(1.6)}{\ln(2)} \].Calculate \( t \).

Step 7: Calculation result

Carrying out the calculation:\[ t \approx 20 \cdot \frac{0.470}{0.693} \approx 13.52 \].Therefore, the population was 8 million after approximately 13.5 years.

Key Concepts

Population Growth

Population growth refers to the change in the number of individuals in a population. It's a key concept in understanding how groups of living organisms increase over time. In the context of cities or countries, this often involves examining how factors such as birth rates, death rates, and migration rates influence the overall population numbers.

When we talk about population growth mathematically, we use models to describe how the population changes. These models can be linear or non-linear. In many real-world scenarios, population growth is better described using exponential models, where growth becomes more rapid in proportion to the growing total number or size of what's being measured. This sort of growth is well captured by the exponential model, which is characterized by a constant growth rate.

In the specific case of Cairo's population, the growth from 5 million to 10 million over 20 years is an example of exponential population growth. Understanding this helps city planners and organizations predict future needs, plan resources, and develop infrastructure accordingly.

Exponential Model

The exponential model is a mathematical way to describe how something grows or decays rapidly over time. The foundation of the exponential model is the equation: \[ y = y_0 e^{kt} \] where:

• \( y \) is the amount at time \( t \),
• \( y_0 \) is the initial amount,
• \( k \) is the growth (or decay) rate,
• and \( t \) is time.

In an exponential model, the rate of growth is directly proportional to the current quantity present. So, as the population increases, the rate at which the population grows also increases.

This model is particularly useful for understanding how populations and other quantities that exhibit "compounding" behavior evolve over time. In our exercise, we used this exponential model to not only calculate the growth rate \( k \) from the known increase in population but also to determine how long it takes for the population to reach a certain size, like 8 million people in Cairo.

Natural Logarithm

The natural logarithm is an important concept when dealing with exponential growth. It is often denoted as \( \ln \) and is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.718. More practically, the natural logarithm helps convert exponential expressions into linear ones, making them easier to manipulate mathematically.

In the context of solving exponential growth problems, the natural logarithm is used to solve for variables in exponential equations. For example, when we rearranged the equation \( 2 = e^{20k} \), we applied the natural logarithm to both sides in order to isolate \( k \). This results in \( \ln(2) = 20k \), simplifying our calculations.

Similarly, to find the time \( t \) when the population reached 8 million, we used the natural logarithm on \( e^{kt} = 1.6 \). This allowed us to find \( t = \frac{\ln(1.6)}{k} \). Understanding how to use the natural logarithm effectively is crucial for interpreting and solving exponential growth problems.