To find the area of a triangle, use the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
First, we calculate the area of the original triangle using its dimensions:
\[ \text{Area}_{\text{original}} = \frac{1}{2} \times 6 , \text{inches} \times 4 , \text{inches} = \frac{1}{2} \times 24 , \text{square inches} = 12 , \text{square inches} \]
Next, when the triangle is dilated by a scale factor of 2.5, the dimensions (height and base) are multiplied by this scale factor. The new height and base measurements are:
\[ \text{New height} = 2.5 \times 4 , \text{inches} = 10 , \text{inches} \] \[ \text{New base} = 2.5 \times 6 , \text{inches} = 15 , \text{inches} \]
Now we can calculate the area of the dilated triangle:
\[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times 15 , \text{inches} \times 10 , \text{inches} = \frac{1}{2} \times 150 , \text{square inches} = 75 , \text{square inches} \]
Thus, the area of Mike's dilated triangle is:
\[ \boxed{75} \]