Problem 117 - College Algebra Essentials

Question

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

So, the ratio of width to height in golden rectangles when the denominator is rationalized and calculated to the nearest hundredth is approximately 1.62.

Step 1: Rationalize the denominator

To rationalize the denominator of the golden ratio \(\frac{2}{\sqrt{5}-1}\), multiply both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{5}+1\). Remember, doing this does not change the value of the ratio, it just changes its form. This gives: \(\frac{2(\sqrt{5}+1)}{(\sqrt{5}-1)(\sqrt{5}+1)} = \frac{2\sqrt{5}+2}{5-1}\)

Step 2: Simplify the expression

Simplify the right hand side to get: \( \frac{2\sqrt{5}+2}{4} = \frac{1}{2}(\sqrt{5}+1) \) which is the golden ratio with rationalized denominator.

Step 3: Evaluate the ratio

Now, evaluate this ratio using a calculator. Set your calculator to the right mode, either degree or radian, according to its specification. Take special note of the decimal place to obtain the value to the nearest hundredth. This gives approximately 1.62.

Key Concepts

Rationalizing Denominators

Rationalizing the denominator of a fraction helps simplify expressions involving roots or radicals. When you have a denominator like \( \sqrt{5} - 1 \), its irrationality makes calculations awkward. To rationalize, multiply both the numerator and denominator by the conjugate of the denominator. For \( \sqrt{5} - 1 \), the conjugate is \( \sqrt{5} + 1 \). This uses the identity

• \((a - b)(a + b) = a^2 - b^2\).

By multiplying, you remove the radical from the denominator:

• \( \frac{2}{\sqrt{5}-1} \times \frac{\sqrt{5}+1}{\sqrt{5}+1} = \frac{2(\sqrt{5}+1)}{(\sqrt{5})^2 - 1^2} \),
• which simplifies to \( \frac{2\sqrt{5} + 2}{4} \).

This process not only simplifies calculations but also keeps the expression equivalent.

Algebraic Expressions

Algebraic expressions involve combining numbers with variables, often using operations like addition, subtraction, multiplication, and division. These expressions can include exponents, radicals, and constants altogether. When working on the golden ratio problem, we deal with the expression \( 2\sqrt{5} + 2 \).

• First, expressions are broken down using algebraic rules.
• Then they are combined to form a new expression, allowing for operations on it.

Algebra makes it possible to simplify, manipulate, and solve expressions like the golden ratio, resulting in a more useful form \( \frac{1}{2}(\sqrt{5}+1) \). Understanding these fundamentals helps in reshaping complex problems into workable forms.

Calculator Use

Using a calculator effectively is vital for evaluating expressions accurately. When calculating the golden ratio to the nearest hundredth, input accuracy is essential:

• Ensure the calculator is in the correct mode, whether it be degree or radian, depending on the functions you'll use.
• Enter the expression carefully. For \( \frac{1}{2}(\sqrt{5}+1) \), key in each part ensuring you respect the operational hierarchy: calculate inside the square root first, then add, then multiply.

After evaluating, check the decimal placement carefully. A calculator allows quick precision, such as converting our expression to approximately 1.62, which represents the ratio of width to height in the golden rectangle.

Simplifying Expressions

Simplifying mathematical expressions makes them easier to understand and work with. After rationalizing the denominator, proceed to simplify the resulting expression, \( \frac{2\sqrt{5} + 2}{4} \). This process involves:

• Breaking down the expression: separating terms.
• Combining like terms: simplifying \( \frac{2\sqrt{5} + 2}{4} \) as \( \frac{1}{2}(\sqrt{5} + 1) \).

Simplified expressions reveal underlying patterns and relationships in problems, making complex formulas or calculations more manageable. They play a crucial role across all mathematical problems, achieving clarity and precision.