Optimizing Collection Efforts

Oct 13, 2025 by wbrum.org 89 views

Introduction

In today's world, recycling is an essential practice that helps conserve natural resources, reduce waste, and mitigate the environmental impact of human activities. Two friends, Marta and Ana, have taken it upon themselves to promote recycling in their community by organizing a school recycling initiative. Marta wants to collect disposable glasses, while Ana aims to gather plastic bottles. In this article, we will delve into the mathematical aspects of their recycling efforts and explore how they can optimize their collection process.

The Problem

Marta collects disposable glasses every 15 days, while Ana adds plastic bottles to the collection every 21 days. The question arises: how can they synchronize their collection efforts to maximize the number of recyclable materials collected? To answer this, we need to analyze the mathematical relationship between their collection frequencies.

Mathematical Analysis

Let's denote the number of disposable glasses collected by Marta as M and the number of plastic bottles collected by Ana as A. Since Marta collects glasses every 15 days, the number of glasses collected in a given time period t can be represented as:

M = t / 15

Similarly, since Ana collects bottles every 21 days, the number of bottles collected in a given time period t can be represented as:

A = t / 21

To find the total number of recyclable materials collected, we need to add the number of glasses and bottles collected:

Total = M + A

Substituting the expressions for M and A, we get:

Total = (t / 15) + (t / 21)

To simplify this expression, we can find the least common multiple (LCM) of 15 and 21, which is 105. Multiplying both terms by 7 (the factor that makes 105 a multiple of both 15 and 21), we get:

Total = (7t / 105) + (7t / 105)

Combining the terms, we get:

Total = (14t / 105)

Optimizing Collection Efforts

Now that we have a mathematical representation of the total number of recyclable materials collected, we can analyze how Marta and Ana can optimize their collection efforts. To maximize the number of recyclable materials collected, they should aim to collect materials as frequently as possible.

Since Marta collects glasses every 15 days, she can collect materials every 7.5 days (half of 15) to maximize her collection frequency. Similarly, Ana can collect bottles every 10.5 days (half of 21) to maximize her collection frequency.

However, since they need to synchronize their collection efforts, they should aim to collect materials at the same frequency. The least common multiple of 7.5 and 10.5 is 30, which means they should aim to collect materials every 30 days.

Conclusion

Recommendations

Based on our analysis, we recommend that Marta and Ana:

Collect disposable glasses every 7.5 days to maximize their collection frequency.
Collect plastic bottles every 10.5 days to maximize their collection frequency.
Synchronize their collection efforts to collect materials every 30 days.

By following these recommendations, Marta and Ana can optimize their recycling initiative and make a positive impact on their community.

Future Directions

This analysis can be extended to other types of recyclable materials, such as paper, cardboard, and glass. Additionally, the mathematical model can be used to analyze the impact of different collection frequencies on the total number of recyclable materials collected.

Limitations

This analysis assumes that Marta and Ana collect materials at the same frequency and that the number of recyclable materials collected is directly proportional to the collection frequency. However, in reality, other factors such as the number of people participating in the collection effort, the availability of collection bins, and the efficiency of the collection process can also impact the total number of recyclable materials collected.

Conclusion

Introduction

In our previous article, we explored the mathematical aspects of Marta and Ana's recycling initiative and provided recommendations for optimizing their collection efforts. In this article, we will address some of the frequently asked questions (FAQs) related to their recycling initiative.

Q: What is the least common multiple (LCM) of 15 and 21?

A: The least common multiple (LCM) of 15 and 21 is 105.

Q: Why did you multiply both terms by 7 to simplify the expression?

A: We multiplied both terms by 7 to make 105 a multiple of both 15 and 21. This allowed us to combine the terms and simplify the expression.

Q: What is the optimal collection frequency for Marta and Ana?

A: The optimal collection frequency for Marta and Ana is every 30 days, which is the least common multiple of 7.5 and 10.5.

Q: Can Marta and Ana collect materials at different frequencies?

A: Yes, Marta and Ana can collect materials at different frequencies, but they should aim to synchronize their efforts to maximize the number of recyclable materials collected.

Q: What are some other factors that can impact the total number of recyclable materials collected?

A: Some other factors that can impact the total number of recyclable materials collected include:

The number of people participating in the collection effort
The availability of collection bins
The efficiency of the collection process

Q: Can the mathematical model be used to analyze the impact of different collection frequencies on the total number of recyclable materials collected?

A: Yes, the mathematical model can be used to analyze the impact of different collection frequencies on the total number of recyclable materials collected.

Q: What are some potential limitations of the mathematical model?

A: Some potential limitations of the mathematical model include:

The assumption that the number of recyclable materials collected is directly proportional to the collection frequency
The assumption that Marta and Ana collect materials at the same frequency
The lack of consideration for other factors that can impact the total number of recyclable materials collected

Q: Can the mathematical model be extended to other types of recyclable materials?

A: Yes, the mathematical model can be extended to other types of recyclable materials, such as paper, cardboard, and glass.

Conclusion

In conclusion, Marta and Ana's recycling initiative presents a mathematical problem that requires careful analysis to optimize their collection efforts. By understanding the mathematical relationship between their collection frequencies, they can synchronize their efforts and maximize the number of recyclable materials collected. We hope that this Q&A article has provided additional insights and answers to some of the frequently asked questions related to their recycling initiative.

Recommendations

Based on our analysis, we recommend that Marta and Ana:

Collect disposable glasses every 7.5 days to maximize their collection frequency.
Collect plastic bottles every 10.5 days to maximize their collection frequency.
Synchronize their collection efforts to collect materials every 30 days.

By following these recommendations, Marta and Ana can optimize their recycling initiative and make a positive impact on their community.

Future Directions

This analysis can be extended to other types of recyclable materials, and the mathematical model can be used to analyze the impact of different collection frequencies on the total number of recyclable materials collected. Additionally, the model can be used to optimize the collection process for other types of recyclable materials.