Problem 73 - Algebra A Combined Function

Question

A golden rectangle is a rectangle whose length is approximately 1.6 times its width. The early Greeks thought that a rectangle with these dimensions was the most pleasing to the eye and examples of the golden rectangle are found in many early works of art. For example, the Parthenon in Athens contains many examples of golden rectangles. Mike Hallahan would like to plant a rectangular garden in the shape of a golden rectangle. If he has 78 feet of fencing available, find the dimensions of the garden.

Step-by-Step Solution

Answer

The dimensions of the garden are 15 feet by 24 feet.

Step 1: Define the Problem

To solve for the dimensions of Mike's garden, we first need to understand that he wants to create a golden rectangle. This means the length ( \( l \) ) is approximately 1.6 times the width ( \( w \) ) of the garden.

Step 2: Create Mathematical Equations

Since the length is 1.6 times the width, we can write the equation: \[ l = 1.6w \]. Additionally, the perimeter of the rectangle (which is given as 78 feet) can be expressed in terms of width and length as:\[ 2l + 2w = 78 \].

Step 3: Substitution

Substitute the expression for \( l \) from Step 2 into the perimeter equation. This gives us:\[ 2(1.6w) + 2w = 78 \].

Step 4: Simplify and Solve for Width

Simplify the equation:\[ 3.2w + 2w = 78 \].Combine like terms:\[ 5.2w = 78 \].Now solve for \( w \):\[ w = \frac{78}{5.2} \approx 15 \].

Step 5: Calculate the Length

Using the width \( w \) calculated above, find the length \( l \) using the golden ratio relationship. \[ l = 1.6 \times 15 = 24 \].

Key Concepts

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around the outside of the rectangle. We calculate it by adding together the lengths of all four sides. For any rectangle, this is found using the formula:

• Perimeter = 2(l + w)

where "l" is the length and "w" is the width.
In the problem about Mike's garden, the total amount of fencing he has is 78 feet. This tells us that the perimeter of his garden needs to be 78 feet.
To find how wide (w) and how long (l) the garden can be, we rearrange the formula to:

• 2l + 2w = 78

This means that the sum of twice the length and twice the width should equal the available fencing, also known as the perimeter.

Golden Ratio

The golden ratio is an irrational number that is approximately 1.618. It appears in various fields such as art, architecture, and nature. This mathematical concept is often denoted by the Greek letter φ (phi).
The golden ratio's importance comes from its unique properties, such as the pleasing proportions it lends to design and structures. In visual art and design, it has been used because it tends to create a sense of balance and harmony.

• The formula for a golden rectangle is l = 1.6w, where l is the length and w is the width.

This formula explains that the length of the rectangle should be approximately 1.6 times greater than its width. This ratio was used in famous ancient structures like the Parthenon.

Rectangular Garden Design

Designing a garden involves considering various factors, such as the shape and size of the space available. A rectangular garden provides versatility and simplicity, making it a common choice.
When Mike decides to use a golden rectangle for his garden, he aims to create an aesthetically pleasing layout. Using the golden ratio helps achieve this, with the length being 1.6 times the width.
This type of design can be practical as well as beautiful. Rectangular gardens can easily be divided into smaller plots, creating sections for different plants or themes. This can maximize function while maintaining beauty. In the exercise, Mike had 78 feet of fencing. By applying the concepts of perimeter and the golden ratio, we found that his garden dimensions should be 15 feet wide and 24 feet long. This allows him to optimize both form and function for his garden.