Problem 63 - Algebra and Trigonometry with Analytic Geometry

Question

The population \(N(t)\) (in millions) of the United States \(t\) years after 1980 may be approximated by the formula \(N(t)=231 e^{0.0103 t}\). When will the population be twice what it was in 1980 ?

Step-by-Step Solution

Answer

The population will be twice what it was in 1980 in the year 2047.

Step 1: Identify the initial population in 1980

Start by determining the population of the United States in 1980. Since the problem states that the population in 1980 can be modeled by the equation \( N(t) = 231 e^{0.0103t} \) with \( t=0 \) for the year 1980, we have: \( N(0) = 231 e^{0.0103 \times 0} = 231 \) million. Thus, the population in 1980 is 231 million.

Step 2: Compute the population goal

Since we want to find when the population is twice that of 1980, the goal population is \( 2 \times 231 = 462 \) million.

Step 3: Set up the equation to solve for t

Use the population growth formula to set up the equation: \( 462 = 231 e^{0.0103t} \). Substitute the goal population into the equation.

Step 4: Isolate the exponential component

Divide both sides of the equation by 231 to isolate the exponential term: \( e^{0.0103t} = \frac{462}{231} = 2 \).

Step 5: Solve for t using logarithms

Apply the natural logarithm to both sides of the equation to solve for \( t \): \( \ln(e^{0.0103t}) = \ln(2) \). This simplifies to \( 0.0103t = \ln(2) \) as \( \ln(e^x) = x \).

Step 6: Calculate t

Finally, solve for \( t \) by dividing both sides by 0.0103: \( t = \frac{\ln(2)}{0.0103} \). Use a calculator to find that \( t \approx 67.28 \).

Step 7: Determine the specific year

Since \( t \) represents the number of years after 1980, add 67 to 1980 to find the specific year: 1980 + 67 = 2047.

Key Concepts

Population Modeling

Population modeling is a mathematical concept used to predict how a population changes over time. It involves using equations to represent population dynamics. These models help us understand growth patterns, such as how quickly a population is increasing.
In the context of the original exercise, the population model is given by the equation:

• \( N(t) = 231 e^{0.0103t} \)

This equation describes the population of the United States, \( N(t) \), in millions as a function of time \( t \) years after 1980. The initial population in 1980, \( N(0) \), is 231 million, as calculated using \( N(0) = 231 e^{0.0103 \times 0} = 231 \).
Exponential functions like this are commonly used in population modeling because they can represent rapid growth seen in populations.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is approximately equal to 2.71828. Natural logarithms are especially useful for solving equations involving exponential growth.
In the original exercise, the natural logarithm is applied to both sides of the equation \(e^{0.0103t} = 2\). This is because the logarithm stands as the inverse of the exponential function:

• \(\ln(e^{x}) = x\).

Therefore, by taking the natural logarithm of both sides, you can bring down the exponent. This simplifies solving the equation for the variable \(t\). In this case, the equation \(\ln(e^{0.0103t}) = \ln(2)\) simplifies to \(0.0103t = \ln(2)\), aiding us in isolating \(t\).

Equation Solving

Equation solving is an essential skill in mathematics that involves finding the value of variables that satisfy the conditions of the equation. In exponential equations, such as the population growth equation in our exercise, isolating the variable involves several steps.
Given the equation \(462 = 231 e^{0.0103t}\):

• First, divide both sides by 231 to simplify it: \(e^{0.0103t} = 2\).
• Next, use the natural logarithm to solve for \(t\): \(\ln(e^{0.0103t}) = \ln(2)\).
• Simplify to get \(0.0103t = \ln(2)\).
• Finally, isolate \(t\) by dividing both sides by 0.0103: \(t = \frac{\ln(2)}{0.0103}\).

These steps illustrate the methodical approach needed to solve exponential equations, helping students grasp how to manipulate such equations effectively.

Growth Rate Calculation

Growth rate calculation is the process of determining the rate at which a population or quantity increases over time. It is a critical component of population modeling, as it helps predict future population sizes.
The growth rate in the given model is represented by the value of 0.0103 in the expression \(e^{0.0103t}\). This value is derived from the exponential growth formula \(N(t) = N_0 e^{rt}\), where:

• \(N_0\) is the initial population (231 million).
• \(r\) is the growth rate (0.0103).
• \(t\) is the time in years after the base year 1980.

To find when the population doubles, you solve \(2 = e^{0.0103t}\). Here, the goal is to determine \(t\), where the population is twice its initial size. Ultimately, it was found to be approximately 67.28 years after 1980, or in the year 2047. This illustrates how understanding and calculating growth rates allow for meaningful predictions about population changes.