Problem 70 - College Algebra

Question

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

Volume function: \( V(x) = 4x^2 - 24x + 32 \).

Step 1: Identify the dimensions of the original rectangle

The rectangle has a width of \( x \) and a length that is twice the width, so the length is \( 2x \). Thus, the original dimensions of the rectangle are \( x \) by \( 2x \).

Step 2: Determine the dimensions after squares are cut

Squares with 2 feet sides are cut from each corner. This reduces both the width and length of the base by 4 feet (2 feet from each side). Thus, the new dimensions are \( x - 4 \) for the width and \( 2x - 4 \) for the length.

Step 3: Identify the height of the box

When the corners are cut out and the sides are folded up, the square cutouts determine the height of the box. Since the squares are each 2 feet on a side, the height of the box is 2 feet.

Step 4: Write the expression for the volume of the box

The volume \( V \) of a box is given by the formula \( V = \text{length} \times \text{width} \times \text{height} \). Substitute the dimensions: \( V = (2x - 4)(x - 4)(2) \).

Step 5: Express the volume as a function of \( x \)

Simplify the expression: \[ V = 2(2x - 4)(x - 4) \]First expand the binomials:\[ (2x - 4)(x - 4) = 2x^2 - 8x - 4x + 16 = 2x^2 - 12x + 16 \]Now multiply by 2:\[ V = 2(2x^2 - 12x + 16) = 4x^2 - 24x + 32 \].Thus, the volume as a function of \( x \) is \( V(x) = 4x^2 - 24x + 32 \).

Key Concepts

Rectangle Dimensions

To solve the problem of finding the volume of the box, it's crucial to first understand the original dimensions of the rectangle. The rectangle's width is denoted by \( x \). Meanwhile, the length is twice the width; thus, it's represented as \( 2x \). This forms the basis of many rectangular dimension problems where one side is defined based on its relation to another.

• Width: \( x \)
• Length: \( 2x \)

These dimensions, \( x \) by \( 2x \), help frame how we approach modifying the shape as we progress toward creating the box.

Folding Technique

The next step in converting the rectangle into a box involves a specific folding technique. Start by removing small squares from each corner of the rectangle. Each square measures 2 feet on every side. After removal:

• The width of the rectangle becomes \( x - 4 \).
• The length reduces to \( 2x - 4 \).

Once these squares are cut out, fold the edges up along these cuts to form the walls of the box. The height of the box corresponds to the side length of the squares, which is 2 feet.

Quadratic Function

With the basic construction of the box laid down, the challenge is to express the volume as a quadratic function of \( x \). The quadratic function is a crucial aspect here, as it accurately models how the box’s volume changes with different values of \( x \).The volume of a box defined through a quadratic function provides: \[ V(x) = 4x^2 - 24x + 32 \]This function results from multiplying out the dimensions and simplifying the terms. Quadratic equations like this one can be useful in seeing how a geometric change, such as altering a dimension, can affect output consistently.

Box Volume Formula

The formula to calculate the volume of a box is straightforward: \[ V = \text{length} \times \text{width} \times \text{height} \]For this exercise, substitute:

• Length: \( 2x - 4 \)
• Width: \( x - 4 \)
• Height: 2 feet

Therefore, the volume becomes:\[ V = 2(2x - 4)(x - 4) = 4x^2 - 24x + 32 \]Through such substitutions and simplifications, the complex real-world volume calculation simplifies nicely into a clean quadratic expression. Understanding and using these steps effectively is key to working with box volume calculations through algebraic expressions.