Problem 52 - Algebra 1: Concepts and Skills

Question

The floor of a room is 14 feet long by 14 feet wide. How many square feet of carpet are needed to cover the floor?

Step-by-Step Solution

Answer

The area of the floor is 196 square feet, so this is the amount of carpet needed to cover the floor.

Step 1 - Given Values

The floor of the room is a square with each side measuring 14 feet.

Step 2 - Applying the Formula for the Area of a Square

Calculate the area by squaring the measurement of one side. The formula for the area of a square is \(Area = side^2\). Thus, the area of the room is \(14^2\) square feet.

Step 3 - Calculating the Area

Determine the number of square feet by multiplying 14 by 14, which equals 196 square feet. So, 196 square feet of carpet is needed to cover the floor.

Key Concepts

Square Footage Calculation

Calculating the square footage of a room is crucial when determining how much material, like carpet, is needed to cover the entire area. In our example, we're dealing with a square-shaped floor. To find the square footage, you need to calculate the area of the floor.

To do this, multiply the length of one side by itself (since all sides of a square are equal) using the formula for the area of a square:

• Area = side × side, or simply, side²
• In our case, side = 14 feet
• Thus, Area = 14 × 14 = 196 square feet

Remember, square footage is important whenever you need to buy floor coverings, paint walls, or even plan new furniture arrangements, so knowing how to calculate it is quite practical.

Geometry

Geometry helps us understand shapes and their properties, and it spans much more than just circles or rectangles. When you're looking at geometric figures like squares, you're applying geometry principles to see how they fit in the real world.

A square, defined and studied in geometry, has some specific properties:

• All four sides are equal
• Every angle is 90 degrees, making it a perfect quadrilateral
• It’s symmetrical, meaning it can be rotated, flipped, and still look the same

Understanding these properties makes it easier to calculate measurements like area or perimeter. In our exercise, knowing that the room is a square simplifies the problem because you only need one measurement to find the area – the length of one side.

Multiplication in Algebra

Multiplication is not just about adding numbers repeatedly. In algebra, it's a key operation used to simplify calculations and solve problems. When you multiply numbers, you're essentially creating a product of these numbers, which in many real-life scenarios, translates into calculating areas or volumes.

In the context of our exercise, we used multiplication to determine how large the area of the floor is:

• Each side of the square floor is 14 feet.
• Multiplying the side by itself (14 × 14) gives us the total area.

This multiplication operation is crucial because it allows us to efficiently calculate how much carpet is needed, offering a clear, numerical insight into spatial dimensions. Mastering multiplication, especially in algebraic terms, provides you with a toolbox for tackling various mathematical and real-world challenges with confidence.