Problem 56
Question
The formula \(A=15.9 e^{0.0235 t}\) models the population of Florida, \(A\), in millions, \(t\) years after 2000 . a. What was the population of Florida in \(2000 ?\) b. When will the population of Florida reach 17.5 million?
Step-by-Step Solution
Verified Answer
The population of Florida in the year 2000 was 15.9 million. The population of Florida will reach 17.5 million around the year 2015.
1Step 1: Determine the Population in 2000
To find out the population in the year 2000, simply substitute \(t = 0\) in the equation, because the year 2000 is our initial time point. This yields: \(A = 15.9 e^{0.0235 * 0} = 15.9\) million.
2Step 2: Set up the Equation for 17.5 Million
To find out when the population will reach 17.5 million, insert \(A = 17.5\) into the equation to get: \(17.5 = 15.9 e^{0.0235t}\).
3Step 3: Solve for the Time
Now rearrange the equation to isolate \(e^{0.0235 t}\) on one side: \(e^{0.0235 t} = 17.5/15.9\). Now, to find \(t\), take the natural logarithm on both sides, which yields: \(0.0235 t = ln(17.5/15.9)\). Finally, to determine \(t\), divide by \(0.0235\), resulting in: \(t = ln(17.5/15.9) / 0.0235\). With the numerical values, this gives approximately: \(t = 14.51\) years from 2000. Therefore, the population reaches 17.5 million around the year 2015.
Key Concepts
Population ModelingExponential FunctionsGrowth Rate Calculation
Population Modeling
Population modeling helps us predict the size of a population over time using mathematical formulas. This concept is crucial for understanding growth trends and planning for the future. In the context of the Florida population exercise, the formula \(A = 15.9 e^{0.0235 t}\) represents how the population of Florida grows exponentially starting from the year 2000.
Population models like this rely on initial data, such as the population at a given start point, and apply a growth rate over time. This particular model uses an exponential function to describe growth, representing real-world scenarios where resources and capacities enable populations to grow at a consistently increasing rate.
This means we can not only estimate the population at a specific time but also evaluate how population dynamics affect things like housing, infrastructure, and resources. Understanding these models helps policymakers make informed decisions about managing and preparing for future population demands.
Population models like this rely on initial data, such as the population at a given start point, and apply a growth rate over time. This particular model uses an exponential function to describe growth, representing real-world scenarios where resources and capacities enable populations to grow at a consistently increasing rate.
This means we can not only estimate the population at a specific time but also evaluate how population dynamics affect things like housing, infrastructure, and resources. Understanding these models helps policymakers make informed decisions about managing and preparing for future population demands.
Exponential Functions
Exponential functions are mathematical expressions of the form \(y = a \cdot e^{bx}\) where \(a\) is a constant, \(b\) is the exponent's coefficient, and \(e\) is the base of the natural logarithm. They are used extensively in real-world contexts to model phenomena that grow at an increasing rate, like populations.
In our population model equation \(A = 15.9 e^{0.0235 t}\), the constant \(15.9\) is the initial population size in millions. The term \(e^{0.0235 t}\) depicts how this population grows exponentially over time \(t\). This model shows that for each year \(t\) after the initial time point, the population increases by a factor related to the growth rate.
Exponential growth leads to rapid increases, making exponential functions essential in fields like biology, finance, and environmental science. Understanding exponential functions helps forecast outcomes, assess impacts, and create strategies for sustainable growth management.
In our population model equation \(A = 15.9 e^{0.0235 t}\), the constant \(15.9\) is the initial population size in millions. The term \(e^{0.0235 t}\) depicts how this population grows exponentially over time \(t\). This model shows that for each year \(t\) after the initial time point, the population increases by a factor related to the growth rate.
Exponential growth leads to rapid increases, making exponential functions essential in fields like biology, finance, and environmental science. Understanding exponential functions helps forecast outcomes, assess impacts, and create strategies for sustainable growth management.
Growth Rate Calculation
Growth rate calculation is central to understanding how quickly a population changes over time. In an exponential model, the growth rate determines how fast the population grows each period.
Using the Florida population model, the growth rate is given by the exponent, 0.0235 in the term \(e^{0.0235 t}\). This value is often expressed as a percentage, so multiplying by 100 gives a growth rate of 2.35% per year. This means the population is expected to grow by 2.35% annually from the starting point.
To calculate when the population reaches a certain size, like 17.5 million, you set up an equation: \(17.5 = 15.9 e^{0.0235 t}\). Solving this requires dividing both sides by 15.9, then using the natural logarithm to isolate \(t\). This calculation shows how math can be used to project future scenarios based on current growth trends. Understanding growth rate calculations helps in anticipating changes and making proactive plans to address future needs.
Using the Florida population model, the growth rate is given by the exponent, 0.0235 in the term \(e^{0.0235 t}\). This value is often expressed as a percentage, so multiplying by 100 gives a growth rate of 2.35% per year. This means the population is expected to grow by 2.35% annually from the starting point.
To calculate when the population reaches a certain size, like 17.5 million, you set up an equation: \(17.5 = 15.9 e^{0.0235 t}\). Solving this requires dividing both sides by 15.9, then using the natural logarithm to isolate \(t\). This calculation shows how math can be used to project future scenarios based on current growth trends. Understanding growth rate calculations helps in anticipating changes and making proactive plans to address future needs.
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