To model the effect of a dilation by a scale factor of 3 on the area of a two-dimensional shape, you can follow these steps:
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Understand the Effect of Dilation: When a two-dimensional shape is dilated by a scale factor \( k \), every linear dimension of the shape (length, width, height, etc.) is multiplied by \( k \). For a scale factor of 3, this means that the length and width of the shape will both be multiplied by 3.
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Area Change Due to Dilation: The area of a two-dimensional shape is proportional to the square of its linear dimensions. Thus, if the original area of the shape is \( A \), and the linear dimensions are scaled by a factor \( k \), the new area \( A' \) after dilation can be calculated as follows: \[ A' = k^2 \cdot A \]
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Apply the Scale Factor: For a scale factor of 3, we calculate the new area as: \[ A' = 3^2 \cdot A = 9A \]
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Conclusion: Therefore, dilating a two-dimensional shape by a scale factor of 3 will increase the area of the shape by a factor of 9. If the original area was, for example, 20 square units, the new area after dilation would be: \[ A' = 9 \cdot 20 = 180 \text{ square units} \]
In summary, dilating a two-dimensional shape by a scale factor of 3 results in an area that is 9 times larger than the original area.