Problem 88
Question
The amount of garbage, \(G,\) produced by a city with population \(p\) is given by \(G=f(p) . G\) is measured in tons per week, and \(p\) is measured in thousands of people. a. The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function \(f\). b. Explain the meaning of the statement \(f(5)=2\).
Step-by-Step Solution
Verified Answer
a. \( f(40) = 13 \).
b. A population of 5,000 produces 2 tons of garbage.
1Step 1: Define Function Given Population and Garbage
The function \( f(p) \) relates the population \( p \) in thousands with the garbage produced \( G \) in tons. For Tola, where the population is 40,000 (which is \( 40 \) in thousands), and produces 13 tons of garbage, we have: \( G = f(40) = 13 \).
2Step 2: Interpret the Given Function Statement
The function statement \( f(5) = 2 \) means that when the city's population is 5,000 (since \( p = 5 \) is in thousands), the city produces 2 tons of garbage per week. This translates the function value at 5 into real-world context.
Key Concepts
Population ModelingGarbage ProductionReal-World ApplicationsMathematical Interpretation
Population Modeling
Population modeling is a mathematical approach to understanding and predicting changes in the size and structure of populations over time. In this exercise, it's about figuring out how different population sizes relate to outcomes, like the amount of garbage produced. By using functions, we can model these complex relationships in a structured way.
Understanding population modeling helps us predict trends and prepare for future challenges.
In this example, the town of Tola uses the function \( f(p) \) to model the relationship between the population, \( p \), and the garbage produced, \( G \). This gives town planners and decision-makers a clear view of how changes in population might affect the town's waste management needs.
Understanding population modeling helps us predict trends and prepare for future challenges.
In this example, the town of Tola uses the function \( f(p) \) to model the relationship between the population, \( p \), and the garbage produced, \( G \). This gives town planners and decision-makers a clear view of how changes in population might affect the town's waste management needs.
Garbage Production
Garbage production is often a direct result of population size. As more people live in a city, the amount of waste generated typically increases. The function \( G = f(p) \) represents this relationship between population \( p \) and garbage \( G \).
In Tola, where there are 40,000 residents, the function is expressed as \( f(40) = 13 \). This means that the population produces 13 tons of garbage weekly.
Understanding garbage production is crucial for effective waste management. It allows cities to allocate resources correctly and implement strategies to reduce waste.
In Tola, where there are 40,000 residents, the function is expressed as \( f(40) = 13 \). This means that the population produces 13 tons of garbage weekly.
Understanding garbage production is crucial for effective waste management. It allows cities to allocate resources correctly and implement strategies to reduce waste.
Real-World Applications
Real-world applications of population and garbage models are vast. They allow cities to make informed decisions related to urban planning and resource allocation.
For instance, Tola's use of \( f(p) \) is a small but powerful example of how cities can anticipate waste management needs. Moreover, it can indicate whether current facilities are adequate or if expansions are necessary.
This mathematical relationship informs policies regarding recycling programs or waste reduction initiatives and helps gauge their effectiveness.
For instance, Tola's use of \( f(p) \) is a small but powerful example of how cities can anticipate waste management needs. Moreover, it can indicate whether current facilities are adequate or if expansions are necessary.
This mathematical relationship informs policies regarding recycling programs or waste reduction initiatives and helps gauge their effectiveness.
Mathematical Interpretation
Mathematical interpretation involves translating real-world problems into mathematical functions and understanding what these functions say about the situation.
In the given exercise, the statement \( f(5) = 2 \) demonstrates that a population of 5,000 produces 2 tons of garbage per week. This function helps to quantify the relationship between population size and garbage output.
By analyzing these functions and deriving meaning from them, communities can better prepare for growth and its environmental impact. This makes sure that mathematical interpretations are a vital part of city planning and infrastructure development.
In the given exercise, the statement \( f(5) = 2 \) demonstrates that a population of 5,000 produces 2 tons of garbage per week. This function helps to quantify the relationship between population size and garbage output.
By analyzing these functions and deriving meaning from them, communities can better prepare for growth and its environmental impact. This makes sure that mathematical interpretations are a vital part of city planning and infrastructure development.
Other exercises in this chapter
Problem 88
Find the composition when \(f(x)=x^{2}+2\) for all \(x \geq 0\) and \(g(x)=\sqrt{x-2}\). $$ (g \circ f)(a) ;(f \circ g)(a) $$
View solution Problem 88
For the following exercises, find the composition when \(f(x)=x^{2}+2\) for all \(x \geq 0\) and \(g(x)=\sqrt{x-2}\). $$(g \circ f)(a) ;(f \circ g)(a)$$
View solution Problem 89
Find the composition when \(f(x)=x^{2}+2\) for all \(x \geq 0\) and \(g(x)=\sqrt{x-2}\). $$ (f \circ g)(11) ;(g \circ f)(11) $$
View solution Problem 89
For the following exercises, find the composition when \(f(x)=x^{2}+2\) for all \(x \geq 0\) and \(g(x)=\sqrt{x-2}\). $$(f \circ g)(11) ;(g \circ f)(11)$$
View solution