Problem 34 - Calculus Early Transcendentals

Question

Suppose that the town of Grayrock had a population of \(10,000\) in 1998 and a population of \(12,000\) in 2003 . Assuming an exponential growth model, in what year will the population reach \(20,000 ?\)

Step-by-Step Solution

Answer

The population will reach 20,000 approximately in the year 2017.

Step 1: Understand Exponential Growth Formula

Exponential growth can be described by the formula \( P(t) = P_0 \cdot e^{kt} \), where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( e \) is the base of the natural logarithm, and \( k \) is the growth rate. We know \( P_0 = 10,000 \) in 1998.

Step 2: Establish Population Points

We have two data points: the population in 1998 was \( P(0) = 10,000 \), and in 2003, which is 5 years later, \( P(5) = 12,000 \). These points will help us find the growth rate \( k \).

Step 3: Write Equation Using Known Data

Using the second data point, the equation becomes \( 12,000 = 10,000 \cdot e^{5k} \). This equation will be used to solve for \( k \).

Step 4: Solve for Growth Rate k

Rearrange the equation \( 12,000 = 10,000 \cdot e^{5k} \) by dividing both sides by \( 10,000 \) to get \( 1.2 = e^{5k} \). Then take the natural logarithm of both sides: \( \ln(1.2) = 5k \). Divide by 5 to find \( k = \frac{\ln(1.2)}{5} \approx 0.0362 \).

Step 5: Set Up Equation for Desired Population

We want to find when the population reaches 20,000. Substitute the values into the original formula: \( 20,000 = 10,000 \cdot e^{0.0362t} \).

Step 6: Solve for Time t

Rearrange the equation \( 20,000 = 10,000 \cdot e^{0.0362t} \) by dividing both sides by \( 10,000 \) to get \( 2 = e^{0.0362t} \). Take the natural logarithm of both sides: \( \ln(2) = 0.0362t \). Finally, solve for \( t \) by dividing \( \ln(2) \) by \( 0.0362 \): \( t \approx 19.16 \).

Step 7: Calculate the Exact Year

Add \( t \approx 19.16 \) years to the base year, 1998. The population will reach 20,000 approximately in the year 1998 + 19 = 2017.

Key Concepts

Growth Rate Calculation

Before diving into predicting future populations using the exponential growth model, we must first calculate the growth rate of the population. This requires identifying the known data points in our situation:

• The initial population (P_0) at a given starting year (e.g., in Grayrock, 1998, with 10,000 people).
• The population at another known time point (e.g., in Grayrock, 2003, when the population was 12,000).

With these data, we apply the formula:\[ 12,000 = 10,000 \cdot e^{5k} \] where the goal is to compute the growth rate (k). Start by isolating \(e^{5k\) on one side:\[ 1.2 = e^{5k} \].

Next, apply the natural logarithm to both sides to eliminate the exponential expression: \( \ln(1.2) = 5k \). Finally, do a simple division to find \(k\), the growth rate:
\[ k = \frac{\ln(1.2)}{5} \approx 0.0362 \].
Hence, the population growth rate for Grayrock is approximately 0.0362 or 3.62% per year.

Exponential Growth Model

The exponential growth model is a mathematical representation used to describe how populations grow non-linearly over time. This model rests on the principle that the growth rate of the population is proportional to its current size. The general formula is as follows:\[ P(t) = P_0 \cdot e^{kt} \] where:

• \(P(t)\) is the future population;
• \(P_0\) is the initial population;
• \(e\) is the base of natural logarithms;
• \(k\) is the growth rate;
• \(t\) is the time elapsed since the initial population measurement.

This model shows us how populations can grow faster and faster because the rate of growth itself increases as the population size increases. It is important to note that while exponential growth models are powerful, they assume an ideal, unlimited resources scenario and may not indefinitely describe long-term growth due to practical constraints.

Population Projection

Once the growth rate has been established using the exponential growth model, our focus shifts to population projection. Predicting future populations with the model involves calculating when the population will hit a certain target.

With our example, and armed with the growth rate \(k = 0.0362\), we use the equation to find when Grayrock's population will meet 20,000:
\[ 20,000 = 10,000 \cdot e^{0.0362t} \].

First, rearrange this equation by dividing both sides by 10,000 to simplify: \(2 = e^{0.0362t}\). Then log-transform both sides:
\(\ln(2) = 0.0362t\).

Finally, compute when this will occur by dividing:\[ t = \frac{\ln(2)}{0.0362} \approx 19.16 \].
This indicates the population will reach 20,000 approximately 19 years after the base year, which is 1998 in our situation. Thus, by 2017, Grayrock is projected to have a population of 20,000, demonstrating the utility of exponential models in forecasting population growth.