Problem 28 - Prealgebra

Question

The Luxor Hotel in Las Vegas is almost an exact model of the pyramid of Khafre, the second largest Egyptian pyramid. The heights of the Luxor hotel and the pyramid of Khafre are 350 feet and 470 feet respectively. If the base of the pyramid in Khafre was 705 feet wide, what is the width of the base of the Luxor Hotel?

Step-by-Step Solution

Answer

The width of the base of the Luxor Hotel is 525 feet.

Step 1: Understand the Problem

We are given two pyramids, the Luxor Hotel and the Pyramid of Khafre. They are similar shapes, which means their corresponding sides are in proportion. We know the heights and the base width of Khafre's pyramid and need to find the width of the base of the Luxor Hotel.

Step 2: Write the Proportion for Similar Shapes

Since the pyramids are similar, the heights of the pyramids and the widths of their bases are proportional. This can be represented by the proportion: \( \frac{\text{Height of Luxor}}{\text{Height of Khafre}} = \frac{\text{Base width of Luxor}}{\text{Base width of Khafre}} \).

Step 3: Substitute Known Values into the Proportion

We have the height of the Luxor Hotel as 350 feet and the height of the Pyramid of Khafre as 470 feet. The base width of Khafre is given as 705 feet. Substitute these values into the proportion: \( \frac{350}{470} = \frac{\text{Base width of Luxor}}{705} \).

Step 4: Solve the Proportion Equation

To find the base width of the Luxor Hotel, \( x \), cross-multiply to solve the equation: \( 350 \times 705 = 470 \times x \). Then, \( x = \frac{350 \times 705}{470} \).

Step 5: Calculate the Base Width of the Luxor Hotel

Compute the value: \( x = \frac{350 \times 705}{470} = 525 \) feet. This is the base width of the Luxor Hotel.

Key Concepts

Understanding Similar Shapes

When we say that two shapes are "similar," we mean they have the same shape but different sizes. Similar shapes have corresponding angles that are equal and sides that are proportional. For example:

• The Luxor Hotel in Las Vegas and the Pyramid of Khafre in Egypt are similar pyramids.
• This means the ratio of their corresponding sides, such as height and base width, are equal.

We use this concept of similarity to solve problems about unknown dimensions. The similarity principle is crucial in geometry, helping us apply known measurements from one shape to find unknown measurements in another.

Mastering Cross-Multiplication

Cross-multiplication is a technique used to solve equations that involve fractions or proportions. It's a handy tool in this problem because it allows us to find an unknown value easily. Here's how it works:

• Set up a proportion, which is two ratios set as equal.
• For example, with our pyramids: \( \frac{350}{470} = \frac{x}{705} \).
• To cross-multiply, multiply the numerator of one fraction by the denominator of the other and set the two products equal.
• In our case, \( 350 \times 705 = 470 \times x \).

With cross-multiplication, we simplify the process of finding "\( x \)," the Luxor Hotel's base width.

Basics of Geometry

Geometry is the branch of mathematics focused on the properties and relations of points, lines, surfaces, and solids. Key principles include:

• Understanding shapes, such as triangles, pyramids, and more.
• Using geometric formulas and theorems to solve problems.
• Applying knowledge of geometry to calculate dimensions, like height or width.

In this case, knowing the geometric properties of similar pyramids allows us to set up and solve a proportion for the Luxor Hotel's base width. Geometry serves as the foundation for these calculations.

Exploring the World of Pyramids

Pyramids are fascinating geometric shapes, often associated with ancient structures. They typically have a polygonal base and triangular sides that converge at a single point (the apex). Here's why they are intriguing:

• The Pyramid of Khafre, one of the most famous pyramids, showcases these geometric properties.
• Modern structures, like the Luxor Hotel, mimic this ancient design.
• Understanding the geometry of pyramids can lead to insights about design and engineering.

Linking history and math, pyramids are more than historical structures; they're perfect examples of geometric concepts like similarity and proportion in action.