Example Question - overlapping squares
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Area Calculation of Overlapping Squares
Let the side of the larger square be \( x \) cm. The area of the larger square is \( x^2 \) cm\(^2\). Given the side of the smaller square is 10 cm, the area of the smaller square is \( 10^2 = 100 \) cm\(^2\). The difference in area between the shaded part B and the unshaded part A is given as 24 cm\(^2\). We know that the unshaded part A is a smaller square with an area of 100 cm\(^2\) minus the part overlapping with the larger square. Let the side of the overlapped square be \( y \) cm. The area of the shaded part B includes two rectangles and the overlapped square: \( 2 \times x \times y + y^2 \). The difference given is: \( 2xy + y^2 - 100 = 24 \). As the side of the larger square is \( x \) and the side of the overlapped square is \( y \), then \( x = 10 + y \). Replace \( x \) with \( 10 + y \) in the difference equation: \( 2(10 + y)y + y^2 - 100 = 24 \) \( 20y + 2y^2 + y^2 - 100 = 24 \) \( 3y^2 + 20y - 124 = 0 \) Solving this quadratic equation by the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 3 \), \( b = 20 \), and \( c = -124 \): \( y = \frac{-20 \pm \sqrt{400 + 1488}}{6} \) \( y = \frac{-20 \pm \sqrt{1888}}{6} \) \( y = \frac{-20 \pm 2\sqrt{472}}{6} \) \( y = \frac{-10 \pm \sqrt{472}}{3} \) Since \( y \) must be positive and it is the smaller dimension, we take the positive root: \( y = \frac{-10 + \sqrt{472}}{3} \) Now we find \( x \): \( x = 10 + y \) \( x = 10 + \frac{-10 + \sqrt{472}}{3} \) The area of the unshaded part A is the side of the smaller square squared: Area of A is \( 10^2 = 100 \) cm\(^2\). The area of the shaded part B is the side of the larger square squared minus the area of A. Using the values of \( x \) and \( y \) found above, we can calculate the exact area of B.
Overlapping Square Areas Problem
<p>Let the side of the larger square be \( x \) cm, therefore its area is \( x^2 \) cm².</p> <p>Let the side of the smaller square be \( (x - 10) \) cm, area is \( (x - 10)^2 \) cm².</p> <p>The area of the shaded part A is the difference between the areas of the two squares: \( x^2 - (x - 10)^2 = 24 \) cm².</p> <p>Solving the equation:</p> <p>\( x^2 - (x^2 - 20x + 100) = 24 \)</p> <p>\( x^2 - x^2 + 20x - 100 = 24 \)</p> <p>\( 20x - 100 = 24 \)</p> <p>\( 20x = 124 \)</p> <p>\( x = 6.2 \) cm</p> <p>The side of the smaller square is \( (6.2 - 10) = -3.8 \) cm, which is not possible.</p> <p>There seems to be an error because the side of the smaller square cannot be a negative number. Please verify the information provided in the question.</p>
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