Problem 70 - College Algebra

Question

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 of the box as a function of the width \(x\) is \(V(x) = 4x^2 - 24x + 32\).

Step 1: Determine Relationships and Dimensions

First, identify the relationships and dimensions based on the problem. Let the width be denoted by \(x\). Since the rectangle is twice as long as it is wide, the length is \(2x\). When 2 feet squares are cut from each corner, the new dimensions are \((x - 4)\) for the width and \((2x - 4)\) for the length, because 2 feet is removed from both sides of each dimension.

Step 2: Account for the New Height

When the squares are cut out and the sides are folded up, the height of the box turns out to be the length of the side of the square cut out, which is 2 feet.

Step 3: Calculate the Volume of the Box

Use the formula for the volume of a box: \[ V = \, \text{length} \, \times \, \text{width} \, \times \, \text{height} \]Substitute the dimensions into the formula: \[ V(x) = (x - 4) \times (2x - 4) \times 2 \]

Step 4: Simplify the Polynomial

Simplify the expression for the volume to write a polynomial function: First, expand the expression \[(x - 4)(2x - 4) = 2x^2 - 4x - 8x + 16 = 2x^2 - 12x + 16\].Then multiply by the height, 2: \[V(x) = 2 \times (2x^2 - 12x + 16) = 4x^2 - 24x + 32\]. Thus, the volume function is \[V(x) = 4x^2 - 24x + 32\].

Key Concepts

Volume of a Box

The volume of a box is a measure of the amount of space it occupies. To find this, we multiply the length, width, and height of the box. Volume is usually expressed in cubic units.

• In this example, we have a box formed by cutting corners from a rectangle and folding the sides.
• The volume helps us understand how much the box can contain.

Calculating this volume required us to transform our dimensions into a polynomial function that reflects these changes.

Rectangular Dimensions

When dealing with a rectangle, understanding its dimensions is key. Rectangles have a length and a width.

• In the problem, the length of the rectangle is twice its width. This is how we determine the initial dimensions.
• If width is denoted as \(x\), then length becomes \(2x\).

Squares are cut from the corners of the rectangle, changing its usable dimensions to make the box.

• This subtraction needs to be properly accounted for, i.e., subtracting from both the length and width accordingly.
• This leads to the expressions \((x - 4)\) for the new width and \((2x - 4)\) for the new length.

These changes set the basis for forming the open-top box.

Polynomial Expression Expansion

Expanding a polynomial involves multiplying out expressions to simplify them into a standard polynomial form. This is essential for finding the volume function of our box.

• For our box, we start with expressions \((x - 4)\) and \((2x - 4)\).

When these expressions are multiplied:

• First distribute the terms: \((x - 4)(2x - 4) = 2x^2 - 4x - 8x + 16\).
• Combine like terms: \(2x^2 - 12x + 16\).

This expanded form, when multiplied by the height, results in the volume polynomial \(4x^2 - 24x + 32\).

Geometry Applications

Geometry helps us solve real-world problems by using shapes and their properties. Here, it's about transforming a 2D rectangle into a 3D box.

• The exercise is a classic demonstration of geometry in action.
• Geometry helped us to determine how cuts affect size and shape.

This knowledge helps us predict how changes to the flat shape impact the resulting box, especially when it comes to volume.

Variable Manipulation

In mathematical problems, variables can change to reflect real-world constraints or measurements. Manipulating these variables to find solutions is crucial.

• Here, the width \(x\) was our main variable, determining all other dimensions.
• Understanding and manipulating \(x\) helped us explore its influence on resulting shapes.

This skill is vital across many math applications. By adjusting \(x\), we could directly calculate its influence on visual and measurable properties like volume.