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
In this problem, we are given the total weight of water bottles carried by Mara to the field, which is represented by the quadratic expression . Our goal is to factorize this expression to determine the number of water bottles and the weight of each water bottle. This problem requires us to apply our knowledge of algebraic expressions and factorization techniques.
Understanding the Problem
Let's break down the problem and understand what is being asked. We are given a quadratic expression that represents the total weight of water bottles. We need to factorize this expression to find the number of water bottles and the weight of each water bottle. This means we need to express the given quadratic expression as a product of two binomial expressions.
Factoring the Quadratic Expression
To factorize the quadratic expression , we need to find two binomial expressions whose product equals the given expression. We can start by looking for common factors or using the factoring method.
Step 1: Look for Common Factors
The first step is to look for common factors in the quadratic expression. We can factor out the greatest common factor (GCF) from the expression.
import sympy as sp
# Define the variable
x = sp.symbols('x')
# Define the quadratic expression
expr = 60*x**2 + 48*x + 24
# Factor out the GCF
gcf = sp.gcd(expr.coeff(x, 2), expr.coeff(x, 1))
factored_expr = gcf * expr / gcf
print(factored_expr)
Step 2: Use the Factoring Method
If there are no common factors, we can use the factoring method to factorize the quadratic expression. This method involves finding two binomial expressions whose product equals the given expression.
# Define the quadratic expression
expr = 60*x**2 + 48*x + 24
# Factor the quadratic expression
factored_expr = sp.factor(expr)
print(factored_expr)
Analyzing the Factored Expression
Once we have factored the quadratic expression, we can analyze the resulting expression to determine the number of water bottles and the weight of each water bottle.
Step 1: Identify the Binomial Expressions
The factored expression is a product of two binomial expressions. We can identify these binomial expressions and analyze their coefficients.
# Define the factored expression
factored_expr = (12*x + 6)*(5*x + 4)
# Print the binomial expressions
print(factored_expr.as_ordered_factors())
Step 2: Determine the Number of Water Bottles
The number of water bottles is represented by the coefficient of the first binomial expression. In this case, the coefficient is 12, which means there are 12 water bottles.
Step 3: Determine the Weight of Each Water Bottle
The weight of each water bottle is represented by the coefficient of the second binomial expression. In this case, the coefficient is 5, which means each water bottle weighs 5 ounces.
Conclusion
In this problem, we factorized the quadratic expression to determine the number of water bottles and the weight of each water bottle. We used the factoring method to factorize the expression and identified the binomial expressions. We then analyzed the coefficients of these expressions to determine the number of water bottles and the weight of each water bottle. This problem requires us to apply our knowledge of algebraic expressions and factorization techniques.
Final Answer
The factorization that represents the number of water bottles and the weight of each water bottle is:
(12x + 6)(5x + 4)
Introduction
In our previous article, we explored the problem of Mara carrying water bottles to the field to share with her team at halftime. We factorized the quadratic expression to determine 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 significance of the quadratic expression in this problem?
A: The quadratic expression represents the total weight of water bottles carried by Mara to the field. It is a mathematical representation of the problem, and we need to factorize it to determine the number of water bottles and the weight of each water bottle.
Q: How do we factorize the quadratic expression ?
A: We can factorize the quadratic expression by looking for common factors or using the factoring method. We can also use algebraic techniques such as grouping or synthetic division to factorize the expression.
Q: What is the difference between the factoring method and the algebraic technique of grouping?
A: The factoring method involves finding two binomial expressions whose product equals the given expression. The algebraic technique of grouping involves rearranging the terms of the expression to facilitate factoring.
Q: How do we determine the number of water bottles from the factored expression?
A: We can determine the number of water bottles by identifying the coefficient of the first binomial expression in the factored expression. In this case, the coefficient is 12, which means there are 12 water bottles.
Q: How do we determine the weight of each water bottle from the factored expression?
A: We can determine the weight of each water bottle by identifying the coefficient of the second binomial expression in the factored expression. In this case, the coefficient is 5, which means each water bottle weighs 5 ounces.
Q: What is the importance of understanding the concept of factorization in mathematics?
A: Understanding the concept of factorization is crucial in mathematics as it allows us to simplify complex expressions and solve problems more efficiently. Factorization is a fundamental concept in algebra and is used extensively in various mathematical applications.
Q: Can you provide an example of a real-world application of factorization?
A: Yes, factorization has numerous real-world applications. For instance, in physics, factorization is used to solve problems related to motion and energy. In engineering, factorization is used to design and optimize systems. In finance, factorization is used to analyze and predict stock prices.
Conclusion
In this article, we answered some frequently asked questions related to the problem of Mara carrying water bottles to the field to share with her team at halftime. We explored the concept of factorization and its importance in mathematics. We also provided examples of real-world applications of factorization.
Final Answer
The factorization that represents the number of water bottles and the weight of each water bottle is:
(12x + 6)(5x + 4)
This factorization indicates that there are 12 water bottles, and each water bottle weighs 5 ounces.