To find the area of the dilated rectangle, we can use the scale factor of the dilation.
The original rectangle has dimensions 3 ft by 2 ft, so its area \( x \) is calculated as follows:
\[ x = \text{length} \times \text{width} = 3 , \text{ft} \times 2 , \text{ft} = 6 , \text{ft}^2. \]
The dilated rectangle has dimensions of 7.5 ft by 5 ft. We can calculate the area of the dilated rectangle:
\[ \text{Area of dilated rectangle} = 7.5 , \text{ft} \times 5 , \text{ft} = 37.5 , \text{ft}^2. \]
Next, we determine the scale factor \( k \) of the dilation. The length increases from 3 ft to 7.5 ft, which can be calculated as:
\[ k = \frac{7.5}{3} = 2.5. \]
The width increases from 2 ft to 5 ft, which can also be calculated as:
\[ k = \frac{5}{2} = 2.5. \]
Since both dimensions increase by the same factor, we can confirm the scale factor is 2.5.
To find the area of the dilated rectangle in terms of the area of the original rectangle \( x \), we apply the dilation factor squared because area scales with the square of the linear dimensions. Thus, the area of the dilated rectangle can be expressed as:
\[ \text{Area of dilated rectangle} = x \cdot k^2 = x \cdot (2.5)^2. \]
So, the expression Jorge can use to discover the area of the dilated rectangle is:
\[ x(2.5)^2. \]
Thus, the correct response is:
x(2.5)².