Jorge draws a rectangle measuring 3 ft . by 2 ff . Then, he dilates the rectangle so it is 7.5 ff . by 5 ff . If x represents the area of the original rectangle, which expression can he use to discover the area of the dilated n \( x+(2.5)^{2} \) \( x(2.5)^{2} \) \( x(2.5) \) \( x(1.5)^{2} \)

To find the area of the dilated rectangle, we first need to calculate the area of the original rectangle. The area \( x \) of the original rectangle is \( 3 \, \text{ft} \times 2 \, \text{ft} = 6 \, \text{ft}^2 \). The dilation factor can be determined by comparing the dimensions: for both width and height, we have \( \frac{7.5 \, \text{ft}}{3 \, \text{ft}} = 2.5 \) and \( \frac{5 \, \text{ft}}{2 \, \text{ft}} = 2.5 \). The area of the dilated rectangle is thus \( (2.5)^2 \) times the area of the original, leading to the expression \( x(2.5)^{2} \). Now, this expression effectively communicates how the transformation of area works with dilation. Each linear dimension scales by a factor, and since area is a two-dimensional measurement, you square that factor for calculating the new area. Dilation is like blowing up a balloon: while the surface remains proportional, the space it occupies, aka area, increases by the square of the dilation factor!