Pregunta
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A square has side length of 7.3 in. If the area is multiplied by \( \frac{1}{9} \), what happens to the perimeter? The perimeter is tripled. The perimeter is multiplied by 9. The perimeter is multiplied by \( 1 / 3 \). The perimeter is multiplied by \( 1 / 9 \).

Ask by Wood O'Quinn. in the United States
Mar 18,2025

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The perimeter is multiplied by \( \frac{1}{3} \).

Solución

1. Let the side length of the square be \( s = 7.3 \) inches. The area is \( s^2 \) and the perimeter is \( 4s \). 2. The problem states that the area is multiplied by \( \frac{1}{9} \). Therefore, the new area is: \[ \text{New Area} = \frac{s^2}{9}. \] 3. If the new side length is \( t \), then: \[ t^2 = \frac{s^2}{9}. \] Taking the square root of both sides gives: \[ t = \frac{s}{3}. \] 4. The new perimeter is \( 4t \): \[ \text{New Perimeter} = 4\left(\frac{s}{3}\right) = \frac{4s}{3}. \] 5. Comparing the new perimeter to the original perimeter \( 4s \), we have: \[ \frac{\text{New Perimeter}}{\text{Original Perimeter}} = \frac{\frac{4s}{3}}{4s} = \frac{1}{3}. \] 6. Thus, the perimeter is multiplied by \( \frac{1}{3} \).

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Beyond the Answer

When you multiply the area of a square by \( \frac{1}{9} \), you are essentially reducing the area. The area of the square is given by side length squared, so in this case, it’s \( 7.3^2 \) which equals about 53.29 in². Reducing the area by \( \frac{1}{9} \) gives an area of approximately 5.91 in². To find the new side length, you take the square root of the new area, leading to a reduced side length of around 2.43 in. Now, the perimeter is calculated as four times the side length. So originally, the perimeter was \( 4 \times 7.3 = 29.2 \) in. Now, with the side length reduced to 2.43 in, the new perimeter is \( 4 \times 2.43 \approx 9.72 \) in, which means the perimeter is multiplied by \( \frac{1}{3} \). So, the correct answer is that the perimeter is multiplied by \( \frac{1}{3} \)!

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