Problem 15 - Precalculus Mathematics for Calculus

Question

These exercises use the population growth model. The population of California was 29.76 million in 1990 and 33.87 million in 2000 . Assume that the population grows exponentially. (a) Find a function that models the population \(t\) years after 1990 . (b) Find the time required for the population to double. (c) Use the function from part (a) to predict the population of California in the year \(2010 .\) Look up California's actual population in \(2010,\) and compare.

Step-by-Step Solution

Answer

(a) \( P(t) = 29.76e^{0.0133t} \); (b) \( t \approx 52.1 \) years; (c) Predicted: 38.3 million; Actual: 37.25 million.

Step 1: Understand Exponential Growth

Exponential growth can be modeled by the function \( P(t) = P_0 e^{rt} \), where \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the time in years since the starting point. Here, \( P_0 = 29.76 \) million.

Step 2: Set Up Two Equations for Known Values

We know \( P(0) = 29.76 \) million and \( P(10) = 33.87 \) million (since 2000 is 10 years after 1990). Substituting into the formula gives us two equations: \( 29.76 = 29.76e^{0} \) and \( 33.87 = 29.76e^{10r} \). The first equation confirms our initial population, and the second allows us to solve for \( r \).

Step 3: Solve for Growth Rate \( r \)

For the second equation, \( 33.87 = 29.76e^{10r} \). Divide both sides by 29.76: \( \frac{33.87}{29.76} = e^{10r} \). Taking the natural logarithm on both sides gives \( \ln\left(\frac{33.87}{29.76}\right) = 10r \). Solving for \( r \), we get \( r = \frac{1}{10} \ln\left(\frac{33.87}{29.76}\right) \approx 0.0133 \).

Step 4: Formulate Population Model

Now that we have \( r \), substitute it back into the exponential growth function: \( P(t) = 29.76e^{0.0133t} \). This is the population model for California.

Step 5: Find Time to Double Population

To find when the population doubles, set \( P(t) = 2P_0 \). Thus, \( 2 \times 29.76 = 29.76e^{0.0133t} \). This simplifies to \( 2 = e^{0.0133t} \). Taking the natural log gives \( \ln(2) = 0.0133t \), so \( t = \frac{\ln(2)}{0.0133} \approx 52.1 \). The population will double in approximately 52 years.

Step 6: Predict Population for 2010

Use the population model to predict the figure for 2010. Since 2010 is 20 years after 1990, substitute \( t = 20 \) into \( P(t) = 29.76e^{0.0133 \times 20} \). Calculating this gives \( P(20) \approx 38.3 \) million.

Step 7: Compare with Actual 2010 Population

The predicted population is approximately 38.3 million. The actual population of California in 2010 was about 37.25 million. Therefore, the model slightly overestimates the actual population.

Key Concepts

Population Model

A population model is a mathematical representation that describes how a population changes over time. In the context of population growth, such models help us understand and predict changes in population sizes based on certain assumptions. An exponential growth model is one common type of population model used when populations grow at a constant percentage rate.

• The initial population ( P_0) is the size of the population at the beginning of the time period being studied.
• The population model predicts the number of individuals in a population after a certain number of years, based on the initial population and the constant growth rate.

These models are valuable for understanding real-world phenomena, such as human population growth, by considering past data to predict future trends. In our exercise, we are tasked with creating such a model for California's population from 1990 onward.

Exponential Function

Exponential functions are crucial in modeling population growth because they describe processes that change proportionally over time. An exponential growth function can be defined as:\[ P(t) = P_0 e^{rt} \]Where:

• P(t)
is the population at time t.
• P_0
is the initial population.
• e is the base of the natural logarithm, approximately equal to 2.718.
• r is the growth rate.
• t is time in years from the start.

Exponential functions are powerful as they demonstrate how quickly populations can grow when subjected to a constant rate. In our scenario, given two population figures at different times, we can use these functions to solve for the unknown growth rate (r). This approach helps in formulating the model used in predicting future population sizes.

Doubling Time

The term "doubling time" refers to the period it takes for a population to double in size at a constant growth rate. It is a key concept in understanding exponential growth. To calculate doubling time, we use the formula:\[ t_d = \frac{\ln(2)}{r} \]Where:

• t_d is the doubling time.
• ln(2) is the natural logarithm of 2, approximately 0.693.
• r refers to the growth rate from the exponential model.

In our exercise, we calculated the doubling time using the growth rate derived from the model. This shows us that with a growth rate of approximately 1.33%, California's population would take about 52 years to double. Understanding doubling time helps in envisaging the long-term implications of current growth trends.

Growth Rate Calculation

Calculating the growth rate in an exponential growth model involves using known past population figures to ascertain the rate at which a population has grown. This is done using the formula related to the exponential function:\[ r = \frac{1}{t} \ln\left(\frac{P(t)}{P_0}\right) \]Where:

• P(t)
is the population at a known future time.
• P_0 is the initial population.
• t is the number of years between the two known population sizes.

The growth rate (r) is a vital parameter in the exponential function, as it defines how fast the population will increase. In our exercise for California, using the given data from 1990 and 2000, we calculated this rate as approximately 0.0133 or 1.33% per year. This calculation is crucial for making accurate predictions in the population model.