Question

Three squares have a total area of 35.25 ππ2 . The larger square has twice the side-length of the middle-sized square. The smaller square has its side length exactly 0.5 cm smaller than the middle-sixed square. Find the side lengths of each of the three squares.

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Answer to a math question Three squares have a total area of 35.25 ππ2 . The larger square has twice the side-length of the middle-sized square. The smaller square has its side length exactly 0.5 cm smaller than the middle-sixed square. Find the side lengths of each of the three squares.

Jayne
4.4
Let's assume the side length of the middle-sized square is x cm. According to the given information, the larger square has twice the side length of the middle-sized square. Therefore, the side length of the larger square is 2x cm. The smaller square has its side length exactly 0.5 cm smaller than the middle-sized square. So, the side length of the smaller square is $x - 0.5$ cm. The area of a square is calculated by squaring its side length. Therefore, we can write the following equations based on the given information: x^2 + $2x$^2 + $x - 0.5$^2 = 35.25 Expanding the equation: x^2 + 4x^2 + x^2 - x + 0.25 = 35.25 Combining like terms: 6x^2 - x + 0.25 = 35.25 Subtracting 35 from both sides: 6x^2 - x - 35 = 0 Now, we have a quadratic equation. We can solve it using various methods, such as factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula: x = $-(-1$ Β± β$(-1$^2 - 4 * 6 * -35)) / $2 * 6$ Simplifying: x = $1 Β± β(1 + 840$) / 12 x = $1 Β± β841$ / 12 x = $1 Β± 29$ / 12 Now, we have two possible values for x: 1. $1 + 29$ / 12 = 30 / 12 = 2.5 2. $1 - 29$ / 12 = -28 / 12 = -2.33 $discard because side lengths cannot be negative$ Therefore, the side length of the middle-sized square is 2.5 cm. The side length of the larger square is twice the side length of the middle-sized square, so it is 2 * 2.5 = 5 cm. The side length of the smaller square is 0.5 cm smaller than the middle-sized square, so it is 2.5 - 0.5 = 2 cm. In summary, the side lengths of the three squares are as follows: Smaller square: 2 cm Middle-sized square: 2.5 cm Larger square: 5 cm
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