Problem 63 - Algebra 1

Question

One room in Jean's apartment is a square measuring 12.2 feet along the base of each wall. How many square feet of wall-to-wall carpet does Jean need to carpet the room?

Step-by-Step Solution

Answer

Jean needs 148.84 square feet of wall-to-wall carpet to carpet the room.

Step 1: Identify the Square Dimensions

Jean's apartment room is a square, which means all sides are equal. Each side is given as 12.2 feet.

Step 2: Calculate the Area of the Square

The area of a square is calculated by squaring the length of one side. So, multiply 12.2 feet by 12.2 feet which gives an area of 148.84 square feet.

Key Concepts

Area of a Square

Calculating the area of a square is one of the foundational skills in geometry. The area represents the amount of space contained within the square. Since all four sides of a square are equal in length, the formula for area is simple: you square the length of one side. This means you multiply the side length by itself. In Jean's case, with a room that has all sides measuring 12.2 feet, you find the area by calculating \(12.2 \text{ feet} \times 12.2 \text{ feet} = 148.84 \text{ square feet}\).

Understanding this concept is crucial because the area of a square is not only a common calculation in everyday situations (like buying carpet for a room as in the given exercise) but also serves as a building block for more complex geometric concepts.

Squaring Numbers

In mathematics, squaring numbers refers to multiplying a number by itself. It is represented by raising a number to the power of two, often written as \(n^2\). When you square a number, you find a value that is exponentially larger. For example, \(5^2 = 5 \times 5 = 25\). When we square a decimal like 12.2, it's no different: \(12.2^2 = 12.2 \times 12.2 = 148.84\).

Squaring numbers isn't just a technique used in geometry; it's also crucial in algebra, where it's involved in forming quadratic equations, and in real-life scenarios, such as determining the area for materials or understanding the relation between different physical quantities.

Dimensional Analysis

The concept of dimensional analysis is a powerful tool used to convert one kind of measurement to another, ensuring that equations make sense dimensionally. It involves units of measurement and is based on the laws of physics that describe the universe. In finding the area of a square, dimensional analysis makes it clear that you're dealing with two dimensions: length and width, both of which are measured in feet in this case. When multiplied, the units themselves are squared, rendering 'square feet' as the result. This ensures that the final answer is not just a number but a quantity that accurately represents the real-world space: \(\text{feet} \times \text{feet} = \text{square feet}\).

Using dimensional analysis helps to prevent errors in calculations across many scientific disciplines and real-world applications, making it a fundamental process in problem-solving.