Problem 70 - College Algebra with Corequisite Support

Question

For the following exercises, use the written statements to construct a polynomial function that represents the required information. A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width \((x)\).

Step-by-Step Solution

Answer

The volume function is \(V(x) = 4x^2 - 32x + 32\).

Step 1: Understanding the problem

We first need to understand the geometry of the open box. The rectangle is described as having a length twice its width, so if the width is \(x\), the length will be \(2x\). Squares with a side of 2 feet are cut from each corner, reducing both the length and width by 4 feet each.

Step 2: Defining dimensions

After the squares are cut out and the sides folded up, the new dimensions of the base of the box become: width: \(x - 4\) and length: \(2x - 4\). The height of the box is the side of the square cut out, which is 2 feet.

Step 3: Constructing the Volume Formula

The volume \(V\) of the box can be expressed as the product of its length \(2x-4\), width \(x-4\), and height 2 (from the squares cut out). Hence, the volume function is given by, \[ V(x) = (2x-4)(x-4)(2) \].

Step 4: Simplifying the Volume Formula

Now we simplify the expression: \[ V(x) = 2(x-4)(2x-4) \]. Distributing the terms, first expand \((x-4)(2x-4)\): \((x \cdot 2x) - (x \cdot 4) - (4 \cdot 2x) + (4 \cdot 4) = 2x^2 - 8x - 8x + 16\), which simplifies to \(2x^2 - 16x + 16\). Finally, multiply by 2: \(V(x) = 2(2x^2 - 16x + 16) = 4x^2 - 32x + 32\).

Key Concepts

Volume Calculation

Volume calculation is an essential concept in geometry and algebra. It helps us determine the amount of space an object occupies. In the context of a box created from a rectangle, understanding how to find its volume becomes practical and straightforward. When shaping the box, we begin by cutting out squares from each corner of the rectangle and folding up the sides to form an open top box. The primary formula for volume is the product of the base area and height. Here, the base area is derived from the modified dimensions after squares are removed:

• Width: \( x - 4 \)
• Length: \( 2x - 4 \)
• Height (from the cut squares): 2 feet

The resulting volume \(V\) of the box can be expressed by multiplying these three dimensions, leading to:\[V(x) = (2x-4)(x-4)(2) = 4x^2 - 32x + 32\] This expression represents the volume as a function of the width \(x\), simplifying our calculations and offering insights into how dimensions change influence volume.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations used to represent real-world problems. In the context of the problem, the volume of the box is represented as a polynomial function of \(x\). This polynomial function, \(V(x) = 4x^2 - 32x + 32\), describes the volume in terms of the box's width, which allows us to compute the volume for various widths quickly. Working with algebraic expressions involves:

• Identifying the variables (here, \(x\) as the width of the rectangle).
• Understanding how changes in \(x\) affect the output (volume).
• Simplifying expressions step-by-step by distributing, combining like terms, and factoring.

As you go through algebraic operations, consistency and clarity are your allies. Each step aligns with previous logic, ensuring that you derive meaningful results. In this exercise, practicing polynomial manipulations sharpens skills essential for solving complex problems.

Geometry Problems

Geometry problems often involve analyzing shapes and their properties, a skill that's particularly evident in constructing our box from a rectangle. Such problems link algebra with physical space, helping us visualize dimensions and transformations. For this problem, the rectangle transforms into an open box. Key steps include:

• Understanding proportional relationships: the rectangle's length is twice its width (\(2x\)).
• Visualizing structural changes by removing equal-sized squares from corners to change the shape.
• Applying geometric principles to identify new dimensions after modifications – reducing both length and width by twice the square's side.

These activities deepen comprehension of geometric principles. Visualizing the rectangle transition into a box through spatial reasoning illustrates both the beauty and the complexity of geometry. By solving such problems, we gain insights into the ways geometrical relationships can be adapted and manipulated for practical applications.