Problem 77 - Algebra 1

Question

Moving The van that you are using to move can hold 16 moving boxes. Each box can hold 60 pounds of books or 15 pounds of clothes. Let \(b\) represent the number of boxes filled with books. Write a function that represents the total weight \(w\) of the boxes in the van.

Step-by-Step Solution

Answer

The function that represents the total weight \(w\) of the boxes in the van is \(w(b) = 45b + 240\)

Step 1: Understand the Problem

The van can carry 16 moving boxes and each box is either filled with books or clothes. For boxes filled with books, each box can hold 60 pounds. Let's represent boxes filled with books by letter \(b\). Given these, the weight contribution by the book boxes is \(60b\). Total boxes are 16, so, boxes filled with clothes would be \(16-b\), because total boxes are either filled with books or clothes. Each clothes box can carry 15 pounds, so these boxes contribute \(15(16-b)\) towards the total weight.

Step 2: Formulate the Equation

The total weight in the van will be the weight of the boxes filled with books plus the weight of the boxes filled with clothes. Working with the details discussed, we formulate our function to be \(w(b) = 60b + 15(16 - b)\). This is how the total weight of the boxes is distributed into weight from book boxes and clothes boxes.

Step 3: Simplify the Function

To further simplify the function \(w(b)\), distribute the 15 in the second term: \(w(b) = 60b + 240 - 15b\). And then combine like terms to get the final function \(w(b) = 45b + 240\)

Key Concepts

Function Representation

Understanding function representation is fundamental in mathematics, especially when working with algebraic functions. In this context, a function describes how the total weight of the boxes is determined by the number of boxes filled with books. To represent this scenario as a function, we focus on defining the relationship between independent and dependent variables. Here, the variable \( b \) represents the number of boxes filled with books. Consequently, the formula that represents the total weight \( w \) of the boxes in the van is a function of \( b \). This function captures all possible combinations of book and clothes boxes within the 16-box limit:

• The part \( 60b \) accounts for the weight contributed by the book boxes.
• The segment \( 15(16-b) \) represents the weight added by the clothes boxes.

Now, combining these segments, we form the expression: \( w(b) = 60b + 15(16-b) \). This is the mathematical representation that ties the number of book boxes to the total weight within the van, making function representation a vital tool to translate real-life problems into usable mathematical solutions.

Problem-Solving

Problem-solving in mathematics often involves understanding the given conditions and translating them into mathematical expressions, as seen in this exercise. Initially, the problem provides specific constraints: the van holds 16 boxes, which can either contain books or clothes. Each type of box contributes differently to the total weight.The essence of solving this problem lies in breaking down the details and constructing a logical pathway to an equation. Starting with known quantities, like the total number of boxes and the weight each type can carry, you deduce:

• The weight contribution by book boxes using \( 60b \).
• The remaining capacity for clothes boxes \( 16-b \), which equates to a weight of \( 15(16-b) \).

By setting up clear equations that reflect these contributions, a comprehensive function \( w(b) = 60b + 15(16-b) \) is formulated. The simplifying step further clarifies this function, ultimately leading to \( w(b) = 45b + 240 \), illustrating the total weight as reliant on the number of book boxes \( b \). Effective problem-solving is about structuring these logical steps clearly and systematically.

Variable Representation

Variable representation is a key concept in algebra, used here to depict how different elements contribute to the total problem. Variables are symbols, usually letters, that stand for values that can change. In our example, the variable \( b \) stands for the number of boxes filled with books.The choice of \( b \) as a variable reflects the focus on finding how varying numbers of book boxes affect the overall weight. Using variables streamlines the process of writing complex relationships in a manageable form. Here’s how they are utilized:

• \( b \) indicates a specific quantity, allowing for equations that show its impact.
• The expression \( 16-b \) shows the remaining boxes for clothes, capturing how one variable provides information about another.

By assigning the letter \( b \) to represent the number of book boxes, you simplify the development of equations. This representation directly informs the structure of the function \( w(b) \), making it a powerful tool to model real-world situations mathematically. Learning to effectively use variables is crucial for solving problems and representing relationships clearly.