Problem 40 - Intermediate Algebra

Question

Quilting. Throughout history, most artists and designers have felt that golden rectangles with a length 1.618 times as long as their width have the most visually attractive shape. A woman is planning to make a quilt in the shape of a golden rectangle. She has exactly 22 feet of a special lace that she plans to sew around the edge of the quilt. What should the length and width of the quilt be? Round both answers up to the nearest hundredth.

Step-by-Step Solution

Answer

Width: 4.20 ft, Length: 6.80 ft.

Step 1: Understanding the Golden Rectangle

A golden rectangle has a length and width where the ratio of the length (L) to the width (W) is 1.618. Therefore, we can express this relationship with the equation:\[ L = 1.618 imes W \]

Step 2: Set Up the Perimeter Equation

The woman has 22 feet of lace to go around the quilt. The perimeter (P) of a rectangle is given by the formula:\[ P = 2L + 2W \]Substitute the given perimeter into the equation:\[ 22 = 2L + 2W \]

Step 3: Express Length in Terms of Width

From the golden rectangle ratio, we already have \( L = 1.618W \). Substitute this into the perimeter equation:\[ 22 = 2(1.618W) + 2W \]

Step 4: Simplify and Solve for Width

Expand and simplify the equation:\[ 22 = 3.236W + 2W \]\[ 22 = 5.236W \]Solve for \( W \) by dividing both sides by 5.236:\[ W = \frac{22}{5.236} \approx 4.20 \]

Step 5: Calculate Length Using Width

Now that you have \( W = 4.20 \), use the golden ratio equation to find the length (L):\[ L = 1.618 \times 4.20 \approx 6.80 \]

Key Concepts

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around the edge of the rectangle. Think of it like walking around the edges once.
Imagine a rectangular path, and the path traveled covers the entire border.
This principle helps us understand how to measure the border's length in mathematical problems like designing a quilt.

To calculate the perimeter, use the formula:

• Perimeter, \( P \), is calculated as follows:

\[ P = 2L + 2W \]where:

• \( L \) is the length of the rectangle, and
• \( W \) is the width of the rectangle.

This formula essentially sums up all sides around the rectangle, doubling the length and width because each of them is repeated twice in a rectangle.

In the context of the quilt problem, knowing that the perimeter is 22 feet helps set up the equations needed to find specific dimensions, ensuring the decorative lace fits perfectly around the quilt's entire perimeter.

Ratio and Proportion

Ratio and proportion are key mathematical concepts used to compare quantities and maintain relationships between dimensions or values. In terms of rectangles, these concepts help determine the relationship between the length and width.

Specifically, the golden ratio, often represented by the Greek letter phi (\( \phi \)), stands at approximately 1.618.
This means for a golden rectangle:

• the ratio of its length (L) to its width (W) should be \( L/W = 1.618 \).

This attractive proportion is considered aesthetically pleasing and can be found in nature, art, and architecture.

In the quilt design exercise, when designing with this ratio, it means if the width is known, the length can be easily calculated using the proportion:

• \( L = 1.618 \, \times \, W \).

This relationship ensures that the quilt maintains the golden rectangle's aesthetically pleasing design.

Quilt Design

Quilt design is a creative and mathematical process that blends art with practical mathematics. It requires a careful arrangement of pieces to form a cohesive and often symmetrical pattern.
In our quilt example, the quilt is designed based on the golden rectangle principles to achieve beauty and functionality.

Quilts often feature repeated patterns and geometric shapes, making it important to understand the mathematical principles underlying their measurements.
For instance, to create a quilt that is a golden rectangle, you start by setting the dimensions where the length is 1.618 times the width, aligning with the principle that such rectangles offer visual harmony.

The lace used to border the quilt symbolizes not only decoration but also the perimeter, which in this case measures 22 feet. Using this input, detailed calculations allow one to determine the exact length and width of the quilt, ensuring all elements fit perfectly, both practically and visually.