Question

Three squares have a total area of 35.25 𝑐𝑚2 . 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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Jayne

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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) ± √((-1)^2 - 4 * 6 * -35)) / (2 * 6)
Simplifying:
x = (1 ± √(1 + 840)) / 12
x = (1 ± √841) / 12
x = (1 ± 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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