First, we can find the area of the original rectangle. The area \( x \) of the original rectangle, which measures 3 ft by 2 ft, is calculated as:
\[ x = 3 \times 2 = 6 \text{ square feet} \]
Next, we calculate the area of the dilated rectangle, which measures 7.5 ft by 5 ft:
\[ \text{Area of the dilated rectangle} = 7.5 \times 5 = 37.5 \text{ square feet} \]
To relate the area of the dilated rectangle to the area of the original rectangle using a dilation factor, we first determine the scale factor. The width goes from 3 ft to 7.5 ft, and the height goes from 2 ft to 5 ft. We can find the dilation scale factor for either dimension:
\[ \text{Scale factor for width} = \frac{7.5}{3} = 2.5 \] \[ \text{Scale factor for height} = \frac{5}{2} = 2.5 \]
Since the scale factors are equal, the dilation factor is 2.5.
When a geometric shape is dilated by a scale factor \( k \), the area changes by a factor of \( k^2 \). Thus, the area of the dilated rectangle can be related to the area of the original rectangle by:
\[ \text{Area of the dilated rectangle} = \text{Area of the original rectangle} \times k^2 \]
Substituting in the scale factor \( k = 2.5 \):
\[ \text{Area of the dilated rectangle} = x \times (2.5)^2 \]
Therefore, the expression that Jorge can use to discover the area of the dilated rectangle is:
\(\boxed{A}\) \( x(2.5)^2 \)
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