Problem 117 - College Algebra

Question

The early Greeks believed that the most pleasing of all rectangles were golden rectangles, whose ratio of width to height is $$\frac{w}{h}=\frac{2}{\sqrt{5}-1}$$ The Parthenon at Athens fits into a golden rectangle once the triangular pediment is reconstructed. (IMAGE CANT COPY) Rationalize the denominator of the golden ratio. Then use a calculator and find the ratio of width to height, correct to the nearest hundredth, in golden rectangles.

Step-by-Step Solution

Answer

The ratio of width to height in golden rectangles is approximately 3.24:1

Step 1: Rationalize the Denominator

In the given ratio \(\frac{w}{h}=\frac{2}{\sqrt{5}-1}\), the denominator of the right-hand side contains a square root. Rationalize it by multiplying numerator and denominator by its conjugate which is \(\sqrt{5} + 1\). So the rationalized form becomes \(\frac{w}{h} = 2(\sqrt{5} + 1)\)

Step 2: Calculate Golden Ratio

Substitute the values in the expression for the golden ratio. This gives \( w:h = 2(\sqrt{5} + 1):1 \). Using a calculator, approximate the golden ratio to the nearest hundredth. This would give something around \( w:h \approx 3.24:1\)

Step 3: Find the Ratio of Width to Height

To find the ratio of width to height, divide the width dimension by the height dimension. Thus, the ratio of width to height is approximately 3.24:1 in golden rectangles.

Key Concepts

Golden Rectangle

Exploring the Golden Rectangle begins with understanding why it has been the pinnacle of aesthetic appeal throughout history, notably, the Greek's Parthenon, which showcases its dimensions. The golden rectangle is a geometrical figure with a specific width to height ratio that makes it visually pleasing, and this unique ratio is based on the golden ratio.

In mathematical terms, the golden ratio, often represented by the Greek letter 'phi', is equal to approximately 1.618. It is an irrational number, meaning it cannot be expressed as an exact fraction. However, to construct or represent a golden rectangle, using a simplified form of the golden ratio is common. The early Greeks believed that the ratio of the width (\(w\)) to the height (\(h\)) of the most aesthetically pleasing rectangles was \(\frac{w}{h}=\frac{2}{\sqrt{5}-1}\). It's by applying this ratio that you could yield a rectangle that is deemed 'golden'.

To visualize this, imagine a rectangle where, if you cut a square off the side, what remains is another, smaller golden rectangle. This self-similarity is one of the fascinating properties that make the golden rectangle a favorite across mathematics, art, and architecture.

Rationalizing the Denominator

When dealing with ratios and fractions in mathematics, particularly within geometric contexts, you may encounter the task of rationalizing the denominator. This mathematical process is essential for simplifying expressions and making calculations easier. In our exercise, this step transforms \(\frac{w}{h}=\frac{2}{\sqrt{5}-1}\) into a form without an irrational number in the denominator.

Rationalizing involves multiplying the fraction by a form of one that will eliminate the radical in the denominator. In this instance, the conjugate of \(\sqrt{5}-1\), which is \(\sqrt{5}+1\), is used. Multiplying both the numerator and the denominator by this conjugate gives us a rationalized version of the golden ratio: \(\frac{w}{h}=2(\sqrt{5}+1)\). This manipulative step is vital across many mathematical concepts because it simplifies the expression and aids in performing further operations or evaluations.

Geometric Figures in Algebra

The intricate relationship between geometry and algebra often unveils itself through geometric figures like the golden rectangle. Algebraic expressions represent the dimensions and properties of these figures, allowing us to engage with geometry on a numerical level.

Geometric figures in algebra often involve variables and constants representing lengths, areas, and ratios. The exercise here encapsulates this relationship by expressing the golden ratio algebraically and undergoing operations to find a more graspable value. By working algebraically with the ratio \(\frac{w}{h}=2(\sqrt{5}+1)\), we can calculate numerical approximations for dimensions that otherwise would be abstract geometrical concepts.

The collaboration of geometric figures and algebra helps us calculate, for example, the width to height ratio of the golden rectangle and extends beyond to more complex configurations. It's through algebra that we can unlock a deeper understanding of the figures that not only define our mathematical reality but also the aesthetics of the world we've built around us.