Problem 42 - Prealgebra

Question

City code requires that a reception hall must allow 4 square feet for each person on the dance floor. The reception hall wants to have a dance floor that is a square and that is large enough for 100 people at a time. What is the length of each side of the dance floor?

Step-by-Step Solution

Answer

The length of each side is 20 feet.

Step 1: Calculate Total Area Required

Since each person requires 4 square feet, for 100 people, the total area required is \(100 \times 4 = 400\) square feet.

Step 2: Determine the Shape of the Dance Floor

The problem states the dance floor is a square. To find the side length of a square, we need to take the square root of the total area.

Step 3: Calculate the Side Length

Calculate the side length of the square by finding the square root of 400. \(\sqrt{400} = 20\).

Step 4: Verify the Solution

Verify by ensuring \(20 \times 20 = 400\) square feet, which matches the required area for 100 people.

Key Concepts

Understanding the Square Root

A square root is a mathematical concept that helps us find a number that, when multiplied by itself, equals a given value. In simpler terms, if you know the area of a square, you can find the length of its sides by calculating the square root of the area. For example, if a square has an area of 400 square feet, determining the square root of 400 will give us the side length of the square. Using the formula \( \sqrt{a} \), where \("a"\) is the area, allows us to easily find the dimension we need. When you calculate \( \sqrt{400} \), the result is 20. This means each side of the square is 20 feet long. Calculating square roots is crucial when working with square-shaped objects in geometry, as it helps you transform an area measurement back into a linear measurement. In everyday life, knowing how to find the square root can help solve various problems such as figuring out the side lengths of squares in real estate planning, architecture, and event organization.

Exploring Geometric Shapes: Focusing on Squares

Geometric shapes are essential in understanding the world around us. One of the most common shapes is the square. A square has four equal sides and four right angles, making it a regular quadrilateral. Its symmetry and simplicity make it unique among other shapes.Why are squares important?

• Squares are prevalent in architecture and design, offering simplicity and balance.
• They represent balanced symmetry, which is often aesthetically pleasing.
• Understanding squares is crucial for calculations involving area, perimeter, and other spatial properties.

To calculate the area of a square, you multiply the length of one side by itself, using the formula \( s \times s = s^2 \). The shape's uniformity makes it an ideal choice for many practical applications, such as constructing buildings and designing dance floors, like in our example.

Approaching Problem Solving

Effective problem solving is a valuable skill that involves breaking down complex tasks into manageable steps. With the example of calculating the side length of a dance floor, we can see how important it is to follow a structured approach: 1. **Understand the Requirements:** First, identify the key elements of the problem, like the need for 4 square feet per person and the capacity for 100 people. 2. **Calculate and Analyze:** Multiply the number of people by the space each needs to find the total area, which is 400 square feet in this case. 3. **Apply Geometry:** Recognize the shape of the dance floor as a square and use the area to find the side length with the square root. 4. **Verify Results:** Double-check your calculations to ensure accuracy, just as we did when verifying that a 20-foot side length correctly satisfies the area requirement of 400 square feet. By approaching problems methodically, you can efficiently arrive at solutions, which is essential in both academic and real-world situations.