To find the area of the dilated rectangle, we can first calculate the area of the original rectangle and establish a relationship with the area of the dilated rectangle.
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Calculate the area of the original rectangle: \[ \text{Area}_{\text{original}} = \text{length} \times \text{width} = 3 , \text{ft} \times 2 , \text{ft} = 6 , \text{ft}^2 \] So, \(x = 6 , \text{ft}^2\).
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Calculate the area of the dilated rectangle: \[ \text{Area}_{\text{dilated}} = \text{length} \times \text{width} = 7.5 , \text{ft} \times 5 , \text{ft} = 37.5 , \text{ft}^2 \]
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To find the scaling factor (dilation factor), we can divide the dimensions of the dilated rectangle by those of the original rectangle:
- The length dilation factor is \( \frac{7.5}{3} = 2.5 \)
- The width dilation factor is \( \frac{5}{2} = 2.5 \)
Since the rectangle is dilated uniformly, the area of the dilated rectangle is related to the area of the original rectangle by the square of the dilation factor: \[ \text{Area}{\text{dilated}} = \text{Area}{\text{original}} \times ( \text{dilation factor} )^2 \] Substituting the values: \[ \text{Area}_{\text{dilated}} = x \times (2.5)^2 \]
Thus, the correct expression Jorge can use to discover the area of the dilated rectangle is: \[ \boxed{D. , x(2.5)^2} \]