Problem 104
Question
The formula \(A=25.1 e^{0.0187 t}\) 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 15 years after 2010, so in the year 2025.
1Step 1: Determining Population in 2010
Substitute the value of \(t=0\) into the equation \(A=25.1 e^{0.0187 t}\), since at \(t = 0\), it represents the year 2010. Therefore, \(A = 25.1 e^{0.0187 \cdot 0} = 25.1 e^{0} = 25.1 \) millions.
2Step 2: Estimating Time to Reach 28 Million
To find out when the population reaches 28 million, we set \(A=28\) and solve for \(t\). Hence, we have \(28=25.1 e^{0.0187 t}\). To extract \(t\), the equation can be rearranged and then use natural logarithm. This will yield \(t = \frac{\ln(\frac{28}{25.1})}{0.0187} = 14.656\) years. Hence, round this to the nearest whole year, so \(t\) is approximately 15 years.
Key Concepts
Population ModelingExponential FunctionsNatural Logarithm
Population Modeling
Understanding population dynamics is critical for planning and resource allocation. In mathematical terms, 'population modeling' employs equations to simulate changes within a population over time, such as birth rates, death rates, and migration.
One of the simplest models is the exponential growth model, which assumes that growth rate is constant and proportional to the current population size. For instance, the population of Texas is represented by the formula \(A=25.1 e^{0.0187 t}\) where \(A\) signifies the population in millions after \(t\) years from 2010.
In this model, the base number, 25.1, refers to the initial population size, and \(0.0187\) indicates the continuous growth rate. By plugging in various values for \(t\), we can predict future population sizes or determine population sizes at specific points in the past, as long as the growth conditions remain consistent.
In the given exercise, by substituting \(t=0\), which corresponds to the year 2010, the initial population is revealed. To improve comprehension, think of the exponential growth model as a way to trace population changes over time on a predictable path, assuming no major interruptions.
One of the simplest models is the exponential growth model, which assumes that growth rate is constant and proportional to the current population size. For instance, the population of Texas is represented by the formula \(A=25.1 e^{0.0187 t}\) where \(A\) signifies the population in millions after \(t\) years from 2010.
In this model, the base number, 25.1, refers to the initial population size, and \(0.0187\) indicates the continuous growth rate. By plugging in various values for \(t\), we can predict future population sizes or determine population sizes at specific points in the past, as long as the growth conditions remain consistent.
In the given exercise, by substituting \(t=0\), which corresponds to the year 2010, the initial population is revealed. To improve comprehension, think of the exponential growth model as a way to trace population changes over time on a predictable path, assuming no major interruptions.
Exponential Functions
Exponential functions are the mathematical backbone of various models illustrating rapid growth or decay. They follow the form \(y=ab^{x}\), where \(a\) is the initial amount, \(b\) is the base growth factor (if greater than one, it indicates growth; if between zero and one, it indicates decay), and \(x\) is the time or the period over which the growth or decay occurs.
In our Texas population example, \(A=25.1 e^{0.0187 t}\) is an exponential function where \(e\) is Euler's number, approximately 2.71828, a constant that is the base of natural logarithms. This population model implies that Texas’s population grows exponentially and the rate of growth is proportional to its current size.
To answer questions such as 'When will the population reach a certain size?' you set the population \(A\) to your desired target and solve for \(t\). As shown in our exercise, finding the time when the population will reach 28 million requires rearranging the equation to solve for \(t\), involving the natural logarithm—an important concept that can unwind the exponential function.
In our Texas population example, \(A=25.1 e^{0.0187 t}\) is an exponential function where \(e\) is Euler's number, approximately 2.71828, a constant that is the base of natural logarithms. This population model implies that Texas’s population grows exponentially and the rate of growth is proportional to its current size.
To answer questions such as 'When will the population reach a certain size?' you set the population \(A\) to your desired target and solve for \(t\). As shown in our exercise, finding the time when the population will reach 28 million requires rearranging the equation to solve for \(t\), involving the natural logarithm—an important concept that can unwind the exponential function.
Natural Logarithm
The natural logarithm, denoted as \(\ln\), is a mathematical function that’s used to determine the time it takes for a quantity to reach a certain level in an exponential growth model. It is essentially the inverse operation of exponentiation with base \(e\), Euler's number.
For instance, if we have an equation \(y=e^{x}\), then the natural logarithm of \(y\), \(\ln(y)\), would give us the exponent \(x\). In our population model, solving for the time \(t\) when Texas's population will reach 28 million involves taking the natural logarithm to isolate \(t\). The equation used is \(\ln(\frac{28}{25.1})/0.0187\), effectively reversing the exponential function.
To improve understanding, remember that while exponential functions model growth by repeated multiplications, natural logarithms allow us to untangle that growth, step by step, back to its origins - the point in time we are looking to discover. This concept is vital not just in population modeling, but in economics, physics, and many areas where change processes are modeled mathematically.
For instance, if we have an equation \(y=e^{x}\), then the natural logarithm of \(y\), \(\ln(y)\), would give us the exponent \(x\). In our population model, solving for the time \(t\) when Texas's population will reach 28 million involves taking the natural logarithm to isolate \(t\). The equation used is \(\ln(\frac{28}{25.1})/0.0187\), effectively reversing the exponential function.
To improve understanding, remember that while exponential functions model growth by repeated multiplications, natural logarithms allow us to untangle that growth, step by step, back to its origins - the point in time we are looking to discover. This concept is vital not just in population modeling, but in economics, physics, and many areas where change processes are modeled mathematically.
Other exercises in this chapter
Problem 103
In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. $$ \log _{4} x=-3 $$
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The formula $$ t=\frac{1}{c}[\ln A-\ln (A-N)] $$ describes the time, \(t,\) in weeks, that it takes to achieve mastery of a portion of a task, where \(A\) is th
View solution Problem 104
In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. $$ \log _{64} x=\frac{2}{3} $$
View solution Problem 105
Describe the product rule for logarithms and give an example.
View solution