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Chapter 3: Problem 73

What is the length of \(x\) if the area of the square is \(169 \mathrm{~m}^2 ?\)

Short Answer

Expert verified
\(x = 13 \, \text{m}\)

Step by step solution

01

Determine the Area Formula

Recall that the area of a square is given by the formula \(\text{Area} = s^2\), where \(s\) represents the side length.
02

Set Up the Equation

Given that the area of the square is \(169 \, \text{m}^2\), we set up the equation \(s^2 = 169\).
03

Solve for the Side Length

To find \(s\), the side length, take the square root of both sides of the equation: \(s = \sqrt{169}\.\)
04

Simplify the Square Root

The square root of \(169\) is \(13\), because \(13 \times 13 = 169\). Thus, \(s = 13 \, \text{m}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Area of a Square
Understanding the area of a square is fundamental in geometry. The area measures the total space within the square's boundaries. We calculate the area of a square using the formula \(\text{Area} = s^2\), where \(s\) is the length of one side of the square. This formula comes from multiplying the side length by itself. For example, if the side length is 4 meters, the area will be \(4 \times 4 = 16 \text{ m}^2\). Knowing this formula helps in various applications, such as determining material quantities in construction.
Square Root
The square root is a key mathematical concept. It answers the question: what number, when multiplied by itself, gives the original number? For example, the square root of 169 is 13, because \(\text{13} \times \text{13} = 169\). We denote the square root using the symbol \(\sqrt{}\). Therefore, \(\sqrt{169} = 13\). Understanding square roots is essential for solving quadratic equations and working with areas.
Algebraic Equations
Algebraic equations are mathematical statements that show the equality between two expressions. In our problem, the equation \(s^2 = 169\) indicates that the square of the side length equals 169. To solve for \(\text{s}\), we need to isolate \(\text{s}\). By taking the square root of both sides, we find \(\text{s} = \sqrt{169} = 13\). Solving algebraic equations involves manipulating the equation to find the value of the unknown variable. It's a vital part of algebra and problem-solving.
Problem Solving Steps
Effective problem solving in mathematics involves clear steps. Let's break down the approach:
  • Identify the formula or principle involved. In this case, it's the area of a square \(\text{Area} = s^2\).
  • Set up the equation based on given information. Here, we know \(s^2 = 169\).
  • Solve for the unknown variable. We take the square root of both sides to get \(s = \sqrt{169}\).
  • Simplify the solution. Finding \(\sqrt{169} = 13\) concludes that the side length is 13 meters.
Following these steps ensures a structured and logical approach to solving mathematical problems.

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