To find the area of Mike's dilated triangle, we can start by calculating the area of the original triangle and then apply the effects of the dilation.
The formula for the area \(A\) of a triangle is:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle:
- Base = 6 inches
- Height = 4 inches
Calculating the area of the original triangle:
\[ A_{\text{original}} = \frac{1}{2} \times 6 \times 4 = \frac{1}{2} \times 24 = 12 \text{ square inches} \]
When a shape is dilated by a scale factor \(k\), the area of the new shape is scaled by a factor of \(k^2\). In this case, the scale factor is 2.5:
\[ k = 2.5 \]
Calculating the area of the dilated triangle:
\[ A_{\text{dilated}} = A_{\text{original}} \times k^2 = 12 \times (2.5)^2 \]
Now calculating \( (2.5)^2 \):
\[ (2.5)^2 = 6.25 \]
Now calculating the area of the dilated triangle:
\[ A_{\text{dilated}} = 12 \times 6.25 = 75 \text{ square inches} \]
Thus, the area of Mike's dilated triangle is \( \boxed{75} \) square inches.
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