Mara Carried Water Bottles To The Field To Share With Her Team At Halftime. The Water Bottles Weighed A Total Of $60x^2 + 48x + 24$ Ounces.Which Factorization Could Represent The Number Of Water Bottles And The Weight Of Each Water Bottle?A.

Introduction

Mara, a dedicated team player, carried water bottles to the field to share with her team at halftime. The weight of the water bottles was a crucial factor in their performance, and Mara wanted to ensure that they had enough to stay hydrated throughout the game. The total weight of the water bottles was given as $60x^2 + 48x + 24$ ounces. In this article, we will explore the possible factorization of this expression, which could represent the number of water bottles and the weight of each water bottle.

The Problem

The problem presents us with a quadratic expression $60x^2 + 48x + 24$ that represents the total weight of the water bottles. We are asked to find a factorization of this expression that could represent the number of water bottles and the weight of each water bottle. This means that we need to find two binomials whose product is equal to the given quadratic expression.

Factoring the Quadratic Expression

To factor the quadratic expression $60x^2 + 48x + 24$, we need to find two binomials whose product is equal to the given expression. We can start by looking for common factors in the expression. In this case, we can factor out a $12$ from each term:

$$60x^2 + 48x + 24 = 12(5x^2 + 4x + 2)$$

Now, we need to factor the quadratic expression inside the parentheses. We can use the factoring method of grouping to factor the expression:

$$5x^2 + 4x + 2 = (5x^2 + 10x) + (-4x + 2)$$

$$= 5x(x + 2) - 2(x + 2)$$

$$= (5x - 2)(x + 2)$$

Therefore, the factorization of the quadratic expression $60x^2 + 48x + 24$ is:

$$60x^2 + 48x + 24 = 12(5x - 2)(x + 2)$$

Interpretation

The factorization $12(5x - 2)(x + 2)$ can be interpreted as follows:

• The factor $12$ represents the number of water bottles.
• The factor $(5x - 2)$ represents the weight of each water bottle in ounces.
• The factor $(x + 2)$ represents the number of water bottles that Mara carried.

Therefore, the factorization $12(5x - 2)(x + 2)$ represents the number of water bottles and the weight of each water bottle.

Conclusion

In this article, we explored the possible factorization of the quadratic expression $60x^2 + 48x + 24$, which represents the total weight of the water bottles. We found that the factorization $12(5x - 2)(x + 2)$ represents the number of water bottles and the weight of each water bottle. This factorization can be used to determine the number of water bottles and the weight of each water bottle, which is essential for Mara's team to stay hydrated throughout the game.

Mathematical Concepts

This problem involves the following mathematical concepts:

• Quadratic expressions: A quadratic expression is a polynomial of degree two, which can be written in the form $ax^2 + bx + c$.
• Factoring: Factoring is the process of expressing a polynomial as a product of simpler polynomials.
• Binomials: A binomial is a polynomial with two terms.
• Grouping: Grouping is a factoring method that involves grouping the terms of a polynomial into two or more groups.

Real-World Applications

This problem has real-world applications in various fields, such as:

• Sports: In sports, hydration is essential for athletes to perform at their best. This problem can be used to determine the number of water bottles and the weight of each water bottle, which is essential for athletes to stay hydrated throughout the game.
• Business: In business, factorization can be used to determine the number of products and the price of each product, which is essential for businesses to make informed decisions.
• Science: In science, factorization can be used to determine the number of particles and the energy of each particle, which is essential for scientists to understand the behavior of particles.

Future Research Directions

This problem has several future research directions, such as:

• Developing new factoring methods: Developing new factoring methods can be used to factor quadratic expressions more efficiently.
• Applying factorization to real-world problems: Applying factorization to real-world problems can be used to determine the number of products and the price of each product, which is essential for businesses to make informed decisions.
• Investigating the properties of quadratic expressions: Investigating the properties of quadratic expressions can be used to understand the behavior of quadratic expressions and develop new mathematical models.
  Mara's Water Bottles: A Mathematical Exploration - Q&A =====================================================

Introduction

In our previous article, we explored the possible factorization of the quadratic expression $60x^2 + 48x + 24$, which represents the total weight of the water bottles. We found that the factorization $12(5x - 2)(x + 2)$ represents the number of water bottles and the weight of each water bottle. In this article, we will answer some frequently asked questions related to this problem.

Q&A

Q: What is the total weight of the water bottles?

A: The total weight of the water bottles is given by the quadratic expression $60x^2 + 48x + 24$.

Q: How many water bottles did Mara carry?

A: Mara carried $12$ water bottles.

Q: What is the weight of each water bottle?

A: The weight of each water bottle is given by the factor $(5x - 2)$.

Q: How can we determine the number of water bottles and the weight of each water bottle?

A: We can determine the number of water bottles and the weight of each water bottle by factoring the quadratic expression $60x^2 + 48x + 24$.

Q: What is the significance of the factor $12$ in the factorization?

A: The factor $12$ represents the number of water bottles that Mara carried.

Q: What is the significance of the factor $(5x - 2)$ in the factorization?

A: The factor $(5x - 2)$ represents the weight of each water bottle in ounces.

Q: What is the significance of the factor $(x + 2)$ in the factorization?

A: The factor $(x + 2)$ represents the number of water bottles that Mara carried.

Q: Can we use this factorization to determine the number of water bottles and the weight of each water bottle for different values of $x$?

A: Yes, we can use this factorization to determine the number of water bottles and the weight of each water bottle for different values of $x$.

Q: How can we apply this factorization to real-world problems?

A: We can apply this factorization to real-world problems such as determining the number of products and the price of each product, which is essential for businesses to make informed decisions.

Q: What are some future research directions related to this problem?

A: Some future research directions related to this problem include developing new factoring methods, applying factorization to real-world problems, and investigating the properties of quadratic expressions.

Conclusion

In this article, we answered some frequently asked questions related to the problem of determining the number of water bottles and the weight of each water bottle. We hope that this article has provided a better understanding of the problem and its significance.

Mathematical Concepts

This problem involves the following mathematical concepts:

• Quadratic expressions: A quadratic expression is a polynomial of degree two, which can be written in the form $ax^2 + bx + c$.
• Factoring: Factoring is the process of expressing a polynomial as a product of simpler polynomials.
• Binomials: A binomial is a polynomial with two terms.
• Grouping: Grouping is a factoring method that involves grouping the terms of a polynomial into two or more groups.

Real-World Applications

This problem has real-world applications in various fields, such as:

• Sports: In sports, hydration is essential for athletes to perform at their best. This problem can be used to determine the number of water bottles and the weight of each water bottle, which is essential for athletes to stay hydrated throughout the game.
• Business: In business, factorization can be used to determine the number of products and the price of each product, which is essential for businesses to make informed decisions.
• Science: In science, factorization can be used to determine the number of particles and the energy of each particle, which is essential for scientists to understand the behavior of particles.

Future Research Directions

This problem has several future research directions, such as:

• Developing new factoring methods: Developing new factoring methods can be used to factor quadratic expressions more efficiently.
• Applying factorization to real-world problems: Applying factorization to real-world problems can be used to determine the number of products and the price of each product, which is essential for businesses to make informed decisions.
• Investigating the properties of quadratic expressions: Investigating the properties of quadratic expressions can be used to understand the behavior of quadratic expressions and develop new mathematical models.