Problem 38 - Prealgebra

Question

Europe's largest town square is the Rynek Glowny located in Krakow, Poland. It covers approximately \(48,400\) square yards. How many feet long is a side of the square?

Step-by-Step Solution

Answer

The side of the square is 660 feet long.

Step 1: Understand Square Area Formula

The area of a square is calculated by squaring the length of one of its sides. If the side length is denoted as \(s\), the area \(A\) can be expressed by the formula: \[ A = s^2 \] Given that the area of the square is 48,400 square yards, we identify \(A = 48,400\) square yards.

Step 2: Solve for Side Length in Yards

To find the side length \(s\) in yards, we take the square root of the area: \[ s = \sqrt{A} = \sqrt{48,400} \]Calculating the square root gives us:\( s = 220 \) yards.

Step 3: Convert Yards to Feet

Since 1 yard equals 3 feet, to convert the side length from yards to feet, multiply by 3:\[ s_{\text{feet}} = 220 \times 3 = 660 \text{ feet} \]Thus, the length of a side of the square in feet is 660 feet.

Key Concepts

Calculating Square Roots

When working with geometry, especially squares, calculating square roots is key to finding dimensions from provided area measurements. The area of a square is always expressed as the length of one side squared. In mathematical terms, if you know the area of the square, you can calculate the side length by taking the square root of that area. For example, say you have a square with an area of 48,400 square yards. To find out the length of one side, you perform the following steps:

• Identify the area, which in this case is 48,400 square yards.
• Use the square root formula: \[ s = \sqrt{48,400} \]
• Calculate the square root, which gives us 220. This means each side of the square is 220 yards long.

Calculating square roots is powerful in geometry since it reverses the Squaring operation—moving from area back to side lengths easily.

Unit Conversion

Understanding unit conversion is crucial when your measurements must meet a specific format. Generally, in problems involving measurements, it is common to need to switch between units. For instance, converting yards to feet. Yards and feet are both units for measuring length, with 1 yard equal to 3 feet. When you have the side length of our square in yards (220 yards) and need it in feet, you simply multiply:

• Start by identifying the known length: 220 yards.
• Since 1 yard equals 3 feet, multiply the yard measurement by 3:\[ s_{\text{feet}} = 220 \times 3 = 660 \text{ feet} \]

This calculation gives you the side length in feet. Learning how to perform such conversions ensures the measurements are suited for each context it may appear, whether you're in construction, design, or solving a math problem on a test.

Geometry Concepts

The foundation of geometry involves understanding shapes and their properties. One fundamental shape is the square. A square is a four-sided polygon, or quadrilateral, with all sides of equal length and all angles measuring 90 degrees. These characteristics simplify calculations involving area and perimeter. The area is often the first primary calculation one will perform. It represents the space within the boundaries of the square, calculated as:\[ A = s^2 \]Where \( A \) is the area, and \( s \) is the length of one side. Understanding these core geometry concepts helps solve more complex problems as well:

• Area Concept: Crucial for finding the space covering a shape, often leading back to understanding the square's side length or perimeter.
• Simple Calculations: Squaring the side length provides the area, and taking a square root reverses that calculation if the area is known.

Mastering the basics of squares supports comprehension in geometry overall, making it easier to learn and solve related real-world problems effectively.