Problem 15 - Algebra and Trigonometry

Question

Population of California 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 popuble. California in the year 2010 . Look up California's actual population in 2010 , and compare.

Step-by-Step Solution

Answer

(a) \( P(t) = 29.76 \cdot e^{0.01327t} \). (b) Time to double: 52.27 years. (c) Predicted 2010 population: 38.86 million.

Step 1: Define the Exponential Growth Function

The population grows exponentially, so we use the function \( P(t) = P_0 \cdot e^{rt} \), where \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the time in years since 1990.

Step 2: Calculate the Growth Rate

We know the population in 1990 is 29.76 million and 33.87 million in 2000. First, set up the equation using the exponential model: \( 33.87 = 29.76 \cdot e^{10r} \). Solve for \( r \) by dividing both sides by 29.76, getting \( e^{10r} = \frac{33.87}{29.76} \). Take the natural logarithm of both sides: \( 10r = \ln\left(\frac{33.87}{29.76}\right) \). Calculate \( r \) as \( r = \frac{1}{10} \cdot \ln\left(\frac{33.87}{29.76}\right) \approx 0.01327 \).

Step 3: Define the Population Function

With \( P_0 = 29.76 \) million and \( r \approx 0.01327 \), the function is \( P(t) = 29.76 \cdot e^{0.01327t} \).

Step 4: Find Time for Population to Double

To find when the population doubles, set \( P(t) = 2 \cdot P_0 = 2 \cdot 29.76 \). Substitute into the model: \( 59.52 = 29.76 \cdot e^{0.01327t} \). Divide by 29.76 to isolate the exponential term: \( e^{0.01327t} = 2 \). Take the natural logarithm: \( 0.01327t = \ln(2) \). Solve for \( t \): \( t = \frac{\ln(2)}{0.01327} \approx 52.27 \) years.

Step 5: Predict Population in 2010

Substitute \( t = 20 \) (since 2010 is 20 years after 1990) into the function: \( P(20) = 29.76 \cdot e^{0.01327 \times 20} \). Calculate \( P(20) \approx 29.76 \cdot e^{0.2654} \approx 38.86 \) million.

Key Concepts

Population Growth

Population growth refers to the increase in the number of individuals in a population. In this example, we're examining the population growth of California from 1990 to 2000. Understanding how populations change over time can provide insights into resource demands, urban development, and environmental impacts.

Key points about population growth:

• Often influenced by birth rates, death rates, immigration, and emigration.
• Can be modeled mathematically to predict future changes.
• In this scenario, we are using an exponential growth model, assuming steady growth over the years.

These models are crucial for creating forecasts, planning infrastructure, and managing natural resources efficiently.

Exponential Functions

Exponential functions form the backbone of modeling population growth that is constant over time. In an exponential function, the rate of change increases proportionally with the current value, leading to faster growth as time progresses.

Understanding the Basics:

• The general form of an exponential growth function is given by \( P(t) = P_0 \cdot e^{rt} \).
• Here, \( P_0 \) is the initial amount—in this case, California's population in 1990.
• The constant \( e \approx 2.718 \) is the base of the natural logarithm, critical in continuous growth calculations.
• \( r \) is the growth rate, determined based on observed data over a specific period.
• \( t \) represents the time passed since the starting point.

Exponential functions depict how populations can quickly grow under conditions where the growth rate remains positive and consistent.

Doubling Time

Doubling time is a useful concept in exponential growth, denoting the period it takes for a quantity to double in size. It's calculated easily using the formula that involves the natural logarithm because of its relationship with exponential growth.

Steps to Calculate Doubling Time:

• Set your growth function to double the initial value (\( 2 \cdot P_0 \)).
• Use the formula: \( 2 = e^{rt} \), solve for \( t \).
• This can be simplified to \( t = \frac{\ln(2)}{r} \).

In our California example, the population doubles in approximately 52.27 years given a growth rate of 0.01327. This measure gives a tangible visibility into how fast population could grow under consistent conditions.

Growth Rate Calculation

Calculating the growth rate is essential for setting up an exponential model, especially if you're predicting changes over time based on historical data.
To calculate the growth rate, we use historical population data and the exponential growth formula. Here’s how:

• First, rearrange the exponential formula: \( P(t) = P_0 \cdot e^{rt} \).
• Plug in known values to: \( P(t) = 33.87 \) million, \( P_0 = 29.76 \) million, and \( t = 10 \) years.
• Isolate \( e^{rt} \) by dividing the future and initial population: \( e^{10r} = \frac{33.87}{29.76} \).
• Take the natural logarithm (ln) of both sides to solve for \( r \).
• Finally, divide by the number of years: \( r = \frac{1}{10} \cdot \ln\left(\frac{33.87}{29.76}\right) \).
• This provides a growth rate of approximately 0.01327.

Understanding how to calculate the growth rate allows for more accurate modeling and forecasting of similar scenarios.