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Geometry and Trigonometry Difficulty: Hard

Square A has side lengths that are $166$ times the side lengths of square B. The area of square A is $k$ times the area of square B. What is the value of $k$ ?

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Explanation

The correct answer is $27,556$ . The area of a square is $s^{2}$ , where $s$ is the side length of the square. Let $x$ represent the length of each side of square B. Substituting $x$ for $s$ in $s^{2}$ yields $x^{2}$ . It follows that the area of square B is $x^{2}$ . It’s given that square A has side lengths that are $166$ times the side lengths of square B. Since $x$ represents the length of each side of square B, the length of each side of square A can be represented by the expression $166 x$ . It follows that the area of square A is $left parenthesis 166 x right parenthesis squared$ , or $27,556 x squared$ . It’s given that the area of square A is $k$ times the area of square B. Since the area of square A is equal to $27,556 x squared$ , and the area of square B is equal to $x^{2}$ , an equation representing the given statement is $27,556 x squared equals k x squared$ . Since $x$ represents the length of each side of square B, the value of $x$ must be positive. Therefore, the value of $x^{2}$ is also positive, so it does not equal $0$ . Dividing by $x^{2}$ on both sides of the equation $27,556 x squared equals k x squared$ yields $27,556 equals k$ . Therefore, the value of $k$ is $27,556$ .