Problem 51 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Golden rectangles are rectangles for which the ratio of the width \(w\) to th length \(\ell\) is equal to the ratio of \(\ell\) to \(\ell+w\). The ratio of the length to the width for these rectangles is called the golden ratio. Find the value of the golden ratio using a rectangle with a width of 1 unit.

Step-by-Step Solution

Answer

The golden ratio, given that the width of the rectangle is 1 unit, is \((1 + \sqrt{5})/2\)

Step 1: Set up the equation

Given that the width of the rectangle \(w\) is 1, the golden rule can be written as the ration of the width to the length is the same as the ratio of the length to the length plus width. Thus, we write the equation as follows: \(w/\ell = \ell/(\ell + w)\)

Step 2: Substitute the value of w and simplify

Now that we have the equation, substitute \(w\) with 1, the equation will become: \(1/\ell = \ell/(\ell + 1)\). To get rid of the fractions, multiply through by \(\ell*(\ell + 1)\), which would give us: \(\ell + 1 = \ell^2\).

Step 3: Solve for \(\ell\)

The equation \(\ell + 1 = \ell^2\) is a quadratic equation and can be solved by rearranging it to the standard form and using the quadratic formula. After arranging it becomes: \(\ell^2 - \ell - 1 = 0\). Using the quadratic formula \((-\b \pm \sqrt{\b^2-4ac} )/2a\), where \(a=1, b=-1, c=-1\), we find that \(\ell = (1 + \sqrt{5})/2\) or \(\ell = (1 - \sqrt{5})/2\). However, only \(\ell = (1 + \sqrt{5})/2\) fits the context of the problem because lengths can't be negative.

Step 4: Compute the golden ratio

The golden ratio is determined by the length to the width which can be represented as \(\ell/w\). We substitute the values of \(\ell\) and \(w(=1)\) into the formula which gives us \((1 + \sqrt{5})/2\) as the golden ratio hence the golden ratio when the width is 1 unit is \((1 + \sqrt{5})/2\)

Key Concepts

Golden Rectangle

Golden rectangles have a unique characteristic where the ratio of their width to their length is the same as the ratio of their length to the combined total of their length and width. This special ratio is known as the golden ratio. Imagine you draw a rectangle. If you keep dividing it into a square and a new smaller rectangle that holds the same property, you’re seeing the magic of the golden rectangle at work.

Golden rectangles appear in art, architecture, and nature, making them intriguing and beautiful shapes. Artists and designers use these rectangles for aesthetically pleasing compositions.

Quadratic Equation

A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In our case, we derived a quadratic equation when we wrote the equation for the golden rectangle: \(\ell^2 - \ell - 1 = 0\).

To solve it, we employed the quadratic formula:

• \(\frac{-b \pm \sqrt{b^2-4ac}}{2a}\)
• Where \(a = 1\), \(b = -1\), and \(c = -1\).

Solving this, we found \(\ell\) equals \(\frac{1 + \sqrt{5}}{2}\) since lengths cannot be negative.

Quadratic equations are fundamental in algebra and help in calculating properties of many geometrical and real-world problems.

Ratio and Proportion

Ratio and proportion denote relationships between numbers and quantities. A ratio shows how many times one number contains another, while a proportion states that two ratios are equal. In the case of the golden rectangle, we equate two proportions:

• The width to length: \(w/\ell\)
• The length to the sum of length and width: \(\ell/(\ell + w)\)

Recognizing such relationships helps maintain balance and symmetry, especially in design and geometry.

Problem Solving

Problem solving is essential for tackling mathematical challenges like finding the value of the golden ratio. It involves breaking down complex problems into manageable steps:

1. **Understanding the Problem:** Comprehend what the problem asks; here, it involves the golden ratio in a rectangle.

2. **Setting up an Equation:** Translate the problem into an equation, allowing for manipulable expressions.

3. **Solving the Equation:** Use appropriate mathematical techniques such as substitution and solving quadratic equations.

4. **Verifying and Concluding:** Finally, check the solution's validity and context relevance.

This structured approach boosts efficiency and accuracy in mathematical inquiries and real-life applications.