Problem 88 - College Algebra with Corequisite Support

Question

produced by a city with population \(p\) is given by \(G=f(p) \cdot G\) 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

Answer

(a) \( f(40) = \frac{13}{40} \); (b) A population of 5,000 produces 10 tons of garbage.

Step 1: Assign Values to Variables

Given that the population of Tola is 40,000 people, we need to express this in the form used by the function. Thus, \( p = 40 \) (since population is measured in thousands). It produces 13 tons of garbage, so \( G = 13 \).

Step 2: Use the Function to Express Garbage Production

The formula for garbage production is given by \( G = f(p) \). Substituting the known values, we have \( 13 = f(40) \cdot 40 \).

Step 3: Solve for the Function f(p)

To find \( f(40) \), we rearrange the formula: \( f(40) = \frac{13}{40} \). This means that for every thousand people, the garbage production coefficient is \( \frac{13}{40} \).

Step 4: Interpret the Meaning of f(5)=2

The statement \( f(5)=2 \) implies that in a city with 5,000 people, each thousand people contribute to 2 tons of garbage production per week. Essentially, the function indicates the amount of garbage produced per thousand persons.

Key Concepts

Population Modeling

Population modeling is a mathematical technique used to understand how different populations grow and change over time. It's often used to project future growth rates, resource needs, and other aspects of community planning. In the case of the town of Tola, population modeling helps determine the amount of garbage produced per week based on the number of residents. To effectively use population models, consider these important aspects:

• Variables: These represent the different factors at play, like the population size in thousands and the weekly garbage production in tons.
• Function Expressions: Functions like \( f(p) \) help relate these variables in a meaningful way, where \( p \) is the population and \( f(p) \) expresses the garbage production per thousand people.
• Real data application: Deriving functions from actual data—such as the 40,000 population and 13 tons of garbage production in Tola—helps in making predictions about future garbage production needs.

Population modeling doesn't just apply to garbage. It can be a valuable tool for forecasting things like school enrollment, hospital capacity, and energy use, all based on changing population sizes. Understandably, mastering population modeling can significantly benefit planners and decision-makers in various sectors.

Real-world Applications in Algebra

Algebra isn't just about solving equations in a classroom setting; it has immense real-world applications. When we use a function like \( G = f(p) \), this is a prime example of algebra in action, relating two variables to analyze real scenarios:

• Function Utilization: Utilizing functions in algebra allows us to model and understand diverse real-world phenomena, from economics to engineering.
• Communicating Mathematical Relationships: With a function, we express relationships clearly. In this case, telling us exactly how population changes affect garbage production, something crucial for municipal planning.
• Problem Solving: By interpreting functions and equations, we solve everyday problems. For example, finding \( f(40) = \frac{13}{40} \) gives a precise tool to measure how garbage production scales with changes in population.

Though algebra might seem abstract at first, its practical uses become apparent through models like these that directly impact community and business decision-making processes. Embracing these applications can turn algebra from a subject of study into a powerful tool for understanding our world.

Garbage Production Analysis

Understanding garbage production is vital for effective waste management and environmental conservation. When analyzing garbage production through algebraic functions like \( G = f(p) \), city planners and environmentalists gain insights into how changes in population impact waste output. Key factors to consider include:

• Per Capita Production: The expression \( f(p) \) represents how much garbage each thousand people generate. This allows cities to plan for waste management infrastructure proportionately to populations.
• Data Utilization: Using historical and projected data to calculate \( f(p) \) ensures accuracy in future estimations. The case study showing \( f(5)=2 \) means each group of 1,000 people in that scenario produces 2 tons of waste weekly. Such interpretations help in developing targeted policies and efficient resource allocation.
• Environmental Impact: Managing garbage production analysis helps mitigate negative environmental influence, lowering landfill input and encouraging sustainable practices.

Garbage production analysis highlights how algebra can contribute to creating cleaner, more sustainable cities by pinpointing necessary changes in waste management approaches as populations grow or decline. Therefore, understanding and applying these analyses is critical for cities aiming to improve their environmental footprint and manage resources effectively.