To find the area of the dilated rectangle, we first calculate the area of the original rectangle and then determine the scale factor of the dilation.
The area \( x \) of the original rectangle (measuring 3 ft by 2 ft) is calculated as follows:
\[ x = \text{length} \times \text{width} = 3 , \text{ft} \times 2 , \text{ft} = 6 , \text{ft}^2. \]
Next, we look at the dimensions of the dilated rectangle (7.5 ft by 5 ft):
\[ \text{Area of dilated rectangle} = 7.5 , \text{ft} \times 5 , \text{ft} = 37.5 , \text{ft}^2. \]
To find how the area changes due to dilation, we determine the scale factor for both the length and width:
The scale factor for the length (3 ft to 7.5 ft) is: \[ \frac{7.5}{3} = 2.5. \]
The scale factor for the width (2 ft to 5 ft) is: \[ \frac{5}{2} = 2.5. \]
Since both dimensions are scaled by the same factor (2.5), we can say the area scales by the square of the scale factor.
Thus, the area of the dilated rectangle in terms of the original area \( x \) can be expressed as: \[ \text{Area of dilated rectangle} = x \times (2.5)^2. \]
Therefore, the correct expression Jorge can use to discover the area of the dilated rectangle is:
\[ \text{x(2.5)²} \]
So the answer is: x(2.5)².
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