Problem 104
Question
The formula \(A=25.1 e^{0.0187t}\) models the population of Texas, \(A,\) in millions, \(t\) years after 2010 . a. What was the population of Texas in \(2010 ?\) b. When will the population of Texas reach 28 million?
Step-by-Step Solution
Verified Answer
a. The population of Texas in 2010 was 25.1 million. b. The population of Texas will reach 28 million in 2016.
1Step 1: Find Population in 2010
Substitute \(t=0\) into the given formula \(A=25.1 e^{0.0187t}\) to find the population in 2010. Because exponential functions \(e^0 = 1\), this gives \(A = 25.1\) million.
2Step 2: Find When Population Reaches 28 Million
Set \(A=28\) in the formula \(A=25.1 e^{0.0187t}\) to find the time \(t\), when the population will reach 28 million. Solve for \(t\) using logarithms: \[28 = 25.1 e^{0.0187t}\] Divide both sides by 25.1: \[1.115338 = e^{0.0187t}\] Taking natural logarithm of both sides: \[\ln(1.115338) = 0.0187t\] Solving for \(t\) gives: \(t = \frac{\ln(1.115338)}{0.0187}\) on computing this gives \(t \approx 5.72\). Hence, after about 6 years from 2010 (somewhere in 2016), the population of Texas will reach 28 million.
Key Concepts
Population ModelingSolving Exponential EquationsNatural Logarithms
Population Modeling
Population modeling involves using mathematical formulas to predict the growth of a population over time. It helps in understanding how populations change and can aid in planning for resources, infrastructure, and services.
In the example problem given, an exponential growth model is used to represent the population of a state, specifically Texas. This type of model is characterized by a constant rate of growth proportional to the population's size, reflected in the equation's exponent, which is a product of time and a growth rate constant.
Population models can vary significantly depending on factors such as initial population size, birth rates, death rates, immigration, and emigration. The mathematical representation of these factors is essential for an accurate and useful model.
The simulation of the population of Texas illustrates this approach, utilizing an initial population figure and an estimated growth rate to forecast the population at a future date or to backtrack and identify the population at a past date. This knowledge is crucial for planners and policy-makers who must anticipate the needs and challenges posed by population changes.
In the example problem given, an exponential growth model is used to represent the population of a state, specifically Texas. This type of model is characterized by a constant rate of growth proportional to the population's size, reflected in the equation's exponent, which is a product of time and a growth rate constant.
Population models can vary significantly depending on factors such as initial population size, birth rates, death rates, immigration, and emigration. The mathematical representation of these factors is essential for an accurate and useful model.
The simulation of the population of Texas illustrates this approach, utilizing an initial population figure and an estimated growth rate to forecast the population at a future date or to backtrack and identify the population at a past date. This knowledge is crucial for planners and policy-makers who must anticipate the needs and challenges posed by population changes.
Solving Exponential Equations
Exponential equations are a type of algebraic equation where the variable appears in the exponent. They are ubiquitous in many scientific areas, including compound interest, radioactive decay, and, as seen in our example, population growth.
Solving exponential equations typically involves isolating the exponential expression and then applying logarithms to both sides of the equation to find the variable's value. In our exercise, we're asked to solve for the time when Texas's population will reach a specific size.
The solution process demonstrates isolating the exponential term and the power of logarithms to 'bring down' the exponent, allowing for the straightforward resolution of the variable. This procedure is a fundamental technique in algebra that is critical when the variable is not readily isolable through simple algebraic manipulation. Mastery of this skill is essential for anyone studying scientific fields or any area that uses exponential growth modeling.
Solving exponential equations typically involves isolating the exponential expression and then applying logarithms to both sides of the equation to find the variable's value. In our exercise, we're asked to solve for the time when Texas's population will reach a specific size.
The solution process demonstrates isolating the exponential term and the power of logarithms to 'bring down' the exponent, allowing for the straightforward resolution of the variable. This procedure is a fundamental technique in algebra that is critical when the variable is not readily isolable through simple algebraic manipulation. Mastery of this skill is essential for anyone studying scientific fields or any area that uses exponential growth modeling.
Natural Logarithms
Natural logarithms are a specific kind of logarithm based on the irrational number 'e', which is approximately 2.71828. The natural logarithm of a number is the power to which 'e' must be raised to obtain that number. It is denoted as \(\ln(x)\).
In many mathematical and scientific contexts, natural logarithms are preferred because of their natural properties when dealing with continuous growth or decay rates. They are intimately related to the concept of 'e' and exponential functions.
In the context of the exercise involving population growth, the natural logarithm is used to solve for the value of the time variable 't'. By taking the natural logarithm of both sides of the equation, we can leverage the inverse properties of logarithms and exponential functions to isolate 't'.
Understanding how to use natural logarithms is pivotal in solving equations that model real-world phenomena, such as population dynamics, where time and growth rates play a significant role. This mathematical tool helps transcend mere numerical calculation into comprehensible and actionable information.
In many mathematical and scientific contexts, natural logarithms are preferred because of their natural properties when dealing with continuous growth or decay rates. They are intimately related to the concept of 'e' and exponential functions.
In the context of the exercise involving population growth, the natural logarithm is used to solve for the value of the time variable 't'. By taking the natural logarithm of both sides of the equation, we can leverage the inverse properties of logarithms and exponential functions to isolate 't'.
Understanding how to use natural logarithms is pivotal in solving equations that model real-world phenomena, such as population dynamics, where time and growth rates play a significant role. This mathematical tool helps transcend mere numerical calculation into comprehensible and actionable information.
Other exercises in this chapter
Problem 104
Write each equation in its equivalent exponential form. Then solve for \(x .\) $$ \log _{64} x=\frac{2}{3} $$
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Evaluate each expression without using a calculator. $$ \log _{3}\left(\log _{7} 7\right) $$
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Describe the product rule for logarithms and give an example.
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