To find the area of the dilated square, we first need to determine the side length of the square after dilation.
The original square has a side length of 3 inches. When dilating using a scale factor of 1.5, we multiply the original side length by the scale factor:
\[ \text{New side length} = \text{Original side length} \times \text{Scale factor} = 3 , \text{inches} \times 1.5 = 4.5 , \text{inches} \]
Next, we calculate the area of the dilated square using the formula for the area of a square:
\[ \text{Area} = \text{side length}^2 \]
Thus, the area of the dilated square is:
\[ \text{Area} = (4.5 , \text{inches})^2 = 20.25 , \text{square inches} \]
Therefore, the area of the dilated square is \( \boxed{20.25} \) square inches.