Problem 100 - Introductory and Intermediate Algebra for College Students 4th

Question

The formula \(A=22.9 e^{0.0183 t}\) models the population of Texas, \(A,\) in millions, \(t\) years after 2005 a. What was the population of Texas in \(2005 ?\) b. When will the population of Texas reach 27 million?

Step-by-Step Solution

Answer

So, the population of Texas in 2005 was 22.9 million and the year the population will reach 27 million is approximately, \(2005 + 7.47 \approx 2012.47\).

Step 1: Find the population of Texas in 2005

The given formula calculates the population for any year after 2005, if we know the number of years after 2005. For the year 2005, \(t = 0\). Substituting \(t = 0\) into the given formula gives \(A = 22.9 e^{0.0183 * 0} = 22.9 e^{0} = 22.9\). Thus, in 2005, the population of Texas was 22.9 million.

Step 2: Find the time when the population of Texas reaches 27 million

To find the time when the population of Texas would reach 27 million, we will substitute \(A = 27\) into the given formula and solve for \(t\). This gives us \(27 = 22.9 e^{0.0183 t}\). To solve for \(t\), we rearrange the equation and use the natural logarithm function. This results in \(t = {ln(27/22.9)}/{0.0183}\). Using a calculator to solve this expression gives \(t \approx 7.47\).

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases at a rate proportional to its current value. In population modeling, this means that the larger the population becomes, the faster it grows. The classical equation for this phenomenon is represented as A = A_0 e^{rt}, where A_0 is the initial amount, r is the growth rate, and t is time.

When we apply this to the state of Texas, the population at any given time t years after 2005 is modeled by the formula A=22.9 e^{0.0183 t}. Here, the initial population A_0 is 22.9 million, and the growth rate r is 0.0183 per year. This formula assumes that the population growth will continue at this constant percent rate over time, an idealization that may not account for variabilities in real-life scenarios like migration rates, birth and death rates, policies, and natural disasters.

Natural Logarithms

Natural logarithms, denoted by ln, are a mathematical tool for dealing with exponential equations. The natural logarithm is the inverse of the exponential function when the base is Euler's number e. An equation of the form y = e^x can be solved for x by taking the natural logarithm of both sides, resulting in x = ln(y).

This technique is crucial in the second part of the exercise where we need to find out when the population of Texas will reach 27 million. The formula 27 = 22.9 e^{0.0183 t} is rearranged to isolate the exponent: e^{0.0183 t} = 27/22.9. Taking the natural logarithm of both sides then allows us to solve for t, showing that the relationship between time and population levels can be derived by applying the properties of logarithms.

Algebraic Modeling

Algebraic modeling involves creating mathematical equations to represent real-world situations. It provides a systematic way to analyze and predict outcomes based on various variables. When working with population projections, an algebraic model like A=22.9 e^{0.0183 t} encapsulates the behavior of a population over time through parameters such as the initial value (population size in a base year) and the growth rate.

In our example, the algebraic modeling of population growth allows us to easily calculate the population for any year after 2005 and to predict future population sizes. This model assumes that conditions remain consistent over time, which is a simplification. In real-world applications, models must often be adjusted to reflect changes in trends, resources, or policies that may affect population growth.