To find the area of the original rectangle, we calculate it as follows:
\[ \text{Area} = \text{length} \times \text{width} = 3 \text{ ft} \times 2 \text{ ft} = 6 \text{ ft}^2 \]
Now, let's denote the area of the original rectangle by \( x \). Since \( x = 6 \text{ ft}^2 \), we need to determine how the area of the dilated rectangle is related to \( x \).
The dimensions of the dilated rectangle are \( 7.5 \text{ ft} \) and \( 5 \text{ ft} \). We can calculate the area of the dilated rectangle:
\[ \text{Area (dilated)} = 7.5 \text{ ft} \times 5 \text{ ft} = 37.5 \text{ ft}^2 \]
Next, we need to find a relationship between the area of the dilated rectangle and the original area \( x \).
To do that, we find the scale factor of the dilation. The width scales from \( 2 \text{ ft} \) to \( 5 \text{ ft} \):
\[ \frac{5}{2} = 2.5 \]
Similarly, the length scales from \( 3 \text{ ft} \) to \( 7.5 \text{ ft} \):
\[ \frac{7.5}{3} = 2.5 \]
Thus, the rectangle is dilated by a factor of \( 2.5 \) in both dimensions. This means the area of the dilated rectangle can be calculated by multiplying the original area \( x \) by the square of the scale factor:
\[ \text{Area (dilated)} = x \times (2.5)^2 = x \times 6.25 \]
However, in the context of the expressions given in the question, to find the expression that represents the area of the dilated rectangle based on the area \( x \), the multiplication would be done using the appropriate scale factor that correctly represents the area.
Considering the options provided, the correct expression should reflect that the area of the dilated rectangle is equal to the area of the original rectangle multiplied by \( (2.5)^2 \):
So we can see that none of the options correctly represent \( x \times 6.25 \). However, since \( 2.5 \times 2.5 = 6.25 \), it seems like the intended concept may have had some error in conveying the exact values.
The closest relevant option would typically be \( O x (2.5)^2 \) if we understand it represents multiplying the area by the square of the dilation factor, but based on typical options provided, that might not clearly be listed.
Thus in light of \( O x (2.5)^2 \) would correctly represent how to find the area of the dilated rectangle from the original area \( x \).