Problem 27 - Prealgebra

Question

Eiffel Tower At the Paris Las Vegas Hotel is a replica of the Eiffel Tower in France. The heights of the tower in Las Vegas and the tower in France are 460 feet and \(1,063\) feet respectively. The base of the Eiffel Tower in France is 410 feet wide. What is the width of the base of the tower in Las Vegas? Round to the nearest foot.

Step-by-Step Solution

Answer

The width of the base of the tower in Las Vegas is approximately 177 feet.

Step 1: Understand the Proportionality

The task is to find the width of the base of the Eiffel Tower replica at the Paris Las Vegas Hotel, which is assumed to be a scaled version of the original tower in France. This means the dimensions should be proportional. Therefore, the ratio of the height to the base width of the tower in Las Vegas should match the ratio of the height to the base width of the original Eiffel Tower.

Step 2: Set Up the Proportion Equation

Let's denote the width of the base of the tower in Las Vegas as \( x \). The ratio of the height of the Las Vegas tower to the width of its base should equal the ratio of the height of the original tower to the width of its base. Thus, the proportion is: \[ \frac{460}{x} = \frac{1063}{410} \]

Step 3: Solve the Proportion

Solve for \( x \) by cross-multiplying: \[ 460 \times 410 = 1063 \times x \] This simplifies to: \[ 188,600 = 1063x \] Next, solve for \( x \) by dividing both sides by 1063: \[ x = \frac{188,600}{1063} \] Calculate \( x \).

Step 4: Calculate and Round the Answer

Upon calculation, \( x \approx 177.4958 \). Since the problem asks us to round to the nearest foot, we round 177.4958 to 177.

Key Concepts

Replica

A replica is a reproduction or a copy of an existing object, often on a different scale. In the case of the Eiffel Tower at the Paris Las Vegas Hotel, it is a smaller replica of the original Eiffel Tower in France. The idea of a replica is to mimic the essential features and details of the original.

When creating replicas in architecture, the original design and proportions are crucial. Replicas are often created with a specific scaling factor to maintain the visual and structural integrity of the original building.

• The Eiffel Tower replica is designed to reflect the original's iconic shape and appearance.
• The Las Vegas version is scaled down, maintaining the same proportions as the French landmark.

Understanding replicas helps in appreciating how architects can manage to keep the feel of the original monument, even when built at a different size.

Proportion Equation

A proportion equation is an expression that states two ratios are equal. It involves four numbers (a, b, c, and d) and follows the form \( \frac{a}{b} = \frac{c}{d} \). This equation is essential in solving problems related to scale, such as finding missing dimensions.

In our exercise, we assume the Las Vegas Eiffel Tower has dimensions proportional to the original in France. To express this mathematically, we set up the proportion equation:

• The height to width ratio of the Las Vegas tower compares to the height to width ratio of the original tower.
• This can be set up as \( \frac{460}{x} = \frac{1063}{410} \) where \( x \) is the unknown width we need to find.

By structuring the problem as a proportion equation, we create a pathway to solve for unknowns effectively.

Scaling Factor

The scaling factor is crucial in transforming an original dimension to a replica or model. It is the ratio used to resize dimensions. In replica building or model making, it ensures that all aspects are adjusted appropriately to maintain proportions.

Calculating the scaling factor involves dividing corresponding measures of the original by the replica. In our example:

• The scaling factor is calculated using the height of the towers: \( \frac{460}{1063} \).
• This factor is applied to all dimensions, such as the width of the base, ensuring it assumes accurate proportions.

Understanding and applying the correct scaling factor is vital to maintaining the harmonious proportions between an original piece and its replica.

Cross-Multiplication

Cross-multiplication is a technique used to solve proportion equations effectively. It involves multiplying the diagonals of the equation and helps find unknown values.

For our proportion equation \( \frac{460}{x} = \frac{1063}{410} \):

• By cross-multiplying, we get the equation \( 460 \times 410 = 1063 \times x \).
• This sets up an equality between the cross products of the equation.
• Simplify to find \( x \), proceeding with \( 188,600 = 1063x \).
• Finally, divide to isolate \( x \): \( x = \frac{188,600}{1063} \).

Cross-multiplication simplifies solving for unknowns by transforming a complex proportion into simple arithmetic operations.