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Chapter 3: Problem 81

What is the length of \(x\) if the square has an area of \(289 \mathrm{~m}^2\) ?

Short Answer

Expert verified
The length of x is 17 m.

Step by step solution

01

Understand the Problem

The problem gives the area of a square and asks for the length of one side of the square, denoted as x.
02

Recall the Formula for the Area of a Square

The area of a square can be found using the formula \( \text{Area} = \text{side length}^2 \), which in this case is \( x^2 \).
03

Set Up the Equation

We know that the area is \( 289 \text{ m}^2 \). Set up the equation \( x^2 = 289 \).
04

Solve for x

To find x, take the square root of both sides of the equation: \[ x = \sqrt{289} \].
05

Calculate the Square Root

The square root of 289 is 17, so \( x = 17 \text{ m} \).

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Key Concepts

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

headline of the respective core concept
Let's dive into the concept of calculating the area of a square. A square is a four-sided shape with all sides of equal length. When calculating the area, you use a simple formula:
  • Area = (side length)²
Since all sides of a square are equal, you just need to know the length of one side to find the area. This squared side length gives you the total area covered by the square.
For example, if a square has each side measuring 4 meters, then the area is: (4 meters)² = 16 m².
This formula is crucial for solving problems involving the area of a square, like the original exercise where you needed to determine the length of the sides given the area of 289 m².
headline of the respective core concept
Taking the square root is a fundamental concept in mathematics. It basically answers the question: 'What number, when multiplied by itself, gives the original number?' The square root of a number is represented by the symbol \(\sqrt{}\).
  • The square root of 9 is 3 because \(3^2 = 9\).
  • The square root of 16 is 4 because \(4^2 = 16\).
In our exercise, we needed to find the length of the side of the square given the area of 289 m². To solve this, we took the square root of 289: \(\sqrt{289} = 17\). This means that each side of the square is 17 meters long.
headline of the respective core concept
Solving equations is a big part of math, especially for standardized tests like the GMAT. An equation is a mathematical statement that shows that two expressions are equal. Solving an equation usually means finding the value of the unknown variable.
In the exercise, we were given the area of a square and needed to find the side length. We started with the equation: \(x^2 = 289\).
To solve for x, we took the square root of both sides: \(x = \sqrt{289}\).
This gave us the solution: \(x = 17\). Understanding how to manipulate and solve equations is essential for accurately answering math problems.

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