To find the area of a rectangle, you multiply its length by its width.
First, let's determine the area of the original rectangle that measures 3 ft by 2 ft:
\[ x = \text{length} \times \text{width} = 3 , \text{ft} \times 2 , \text{ft} = 6 , \text{ft}^2. \]
Next, we calculate the area of the dilated rectangle that measures 7.5 ft by 5 ft:
\[ \text{Area of dilated rectangle} = \text{length} \times \text{width} = 7.5 , \text{ft} \times 5 , \text{ft} = 37.5 , \text{ft}^2. \]
To relate the areas, Jorge can express the area of the dilated rectangle in terms of the area of the original rectangle. If \( x \) represents the area of the original rectangle (\( x = 6 , \text{ft}^2 \)), we can express the area of the dilated rectangle as a multiple of the area of the original rectangle.
The scale factor of the dilation can be determined by comparing the dimensions of the rectangles:
- The length scale factor: \( \frac{7.5 , \text{ft}}{3 , \text{ft}} = 2.5 \)
- The width scale factor: \( \frac{5 , \text{ft}}{2 , \text{ft}} = 2.5 \)
Since both dimensions have the same scale factor, we confirm that the rectangle was uniformly dilated by a factor of 2.5.
The area scale factor is the square of the linear scale factor:
\[ \text{Area scale factor} = (2.5)^2 = 6.25. \]
Therefore, the area of the dilated rectangle can be expressed in terms of the original rectangle's area \( x \):
\[ \text{Area of dilated rectangle} = 6.25 \times x. \]
Thus, the expression Jorge can use to discover the area of the dilated rectangle is:
\[ 6.25x. \]