Problem 11
Question
A city's population was 30,700 in the year 2010 and is growing by 850 people a year. (a) Give a formula for the city's population, \(P,\) as a function of the number of years, \(t,\) since \(2010 .\) (b) What is the population predicted to be in \(2020 ?\) (c) When is the population expected to reach \(45,000 ?\)
Step-by-Step Solution
Verified Answer
(a) \[ P(t) = 30,700 + 850t \], (b) 39,200 in 2020, (c) in 2027.
1Step 1: Understand the problem
We are given an initial population for the year 2010 and a constant growth rate per year. We need to find a formula for the population based on the number of years since 2010, predict the population in 2020, and find when the population will reach 45,000.
2Step 2: Establish the formula for population
The population in 2010 is 30,700. Each year, the population grows by 850 people. For any year, the population can be represented as:\[ P(t) = 30,700 + 850t \]where \( t \) is the number of years since 2010.
3Step 3: Calculate population in 2020
To find the population in 2020, calculate \( t = 2020 - 2010 = 10 \). Substitute \( t = 10 \) into the formula:\[ P(10) = 30,700 + 850 \times 10 = 30,700 + 8,500 = 39,200 \]Thus, the population in 2020 is predicted to be 39,200.
4Step 4: Find when population reaches 45,000
Set \( P(t) = 45,000 \) and solve for \( t \):\[ 30,700 + 850t = 45,000 \]Subtract 30,700 from both sides:\[ 850t = 14,300 \]Divide both sides by 850:\[ t = \frac{14,300}{850} = 16.82 \]Since \( t \) must be a whole number of years, round up to 17. Thus, the population is expected to reach 45,000 after 17 years, in 2010 + 17 = 2027.
Key Concepts
Population GrowthApplied CalculusMathematical Modeling
Population Growth
Population growth is an essential area of study in demographics and environmental science. In a city, population growth can significantly impact resources, infrastructure, and the environment. When discussing population growth, we refer to how the number of individuals in a population increases over time.
In this case, we have a linear growth model, meaning the population increases by a constant number each year. Here, the city's population increases by 850 people annually. Such a model is simplified compared to real-world scenarios where factors like birth rates, death rates, immigration, and policy changes could influence growth. However, linear models provide a clear, initial understanding of growth trends.
With an initial population of 30,700 in 2010 and a steady increase, predictions become straightforward. By applying this model, we can anticipate how population changes over a given timeframe.
In this case, we have a linear growth model, meaning the population increases by a constant number each year. Here, the city's population increases by 850 people annually. Such a model is simplified compared to real-world scenarios where factors like birth rates, death rates, immigration, and policy changes could influence growth. However, linear models provide a clear, initial understanding of growth trends.
With an initial population of 30,700 in 2010 and a steady increase, predictions become straightforward. By applying this model, we can anticipate how population changes over a given timeframe.
Applied Calculus
Applied calculus involves using mathematical concepts and techniques to solve real-world problems. In population growth modeling, calculus may be used to understand changes and dynamics over continuous time.
In this exercise, however, the growth is simplified to a linear form, which doesn't typically require advanced calculus, but instead uses basic algebraic functions. The formula for population growth, \(P(t) = 30,700 + 850t\), is an application of simple linear equations. Here, \(P(t)\) represents the population depending on \(t\), the number of years since 2010.
In more complex scenarios, calculus can help analyze non-linear growth, where factors influencing populations change dynamically. Calculus allows us to find rates of change at any given moment and model these changes accurately, which is invaluable in more sophisticated population studies.
In this exercise, however, the growth is simplified to a linear form, which doesn't typically require advanced calculus, but instead uses basic algebraic functions. The formula for population growth, \(P(t) = 30,700 + 850t\), is an application of simple linear equations. Here, \(P(t)\) represents the population depending on \(t\), the number of years since 2010.
In more complex scenarios, calculus can help analyze non-linear growth, where factors influencing populations change dynamically. Calculus allows us to find rates of change at any given moment and model these changes accurately, which is invaluable in more sophisticated population studies.
Mathematical Modeling
Mathematical modeling is a process of representing real-world situations using mathematical language and structures. It provides a way to interpret and predict behaviors and outcomes using mathematical expressions.
For this exercise, the linear function \(P(t) = 30,700 + 850t\) is the model representing the city's population over time. This model helps to predict future population sizes and understand how changes happen over time.
Models are invaluable because they transform abstract concepts into concrete, manageable forms that can be analyzed and tested. The linear model's simplicity allows for easy calculations and predictions. However, it's crucial to remember that all models are simplifications. Despite being simple, this model effectively shows basic growth trends and offers a starting point for more detailed analysis, should additional data become available.
For this exercise, the linear function \(P(t) = 30,700 + 850t\) is the model representing the city's population over time. This model helps to predict future population sizes and understand how changes happen over time.
Models are invaluable because they transform abstract concepts into concrete, manageable forms that can be analyzed and tested. The linear model's simplicity allows for easy calculations and predictions. However, it's crucial to remember that all models are simplifications. Despite being simple, this model effectively shows basic growth trends and offers a starting point for more detailed analysis, should additional data become available.
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Problem 11
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