To find the area of the dilated triangle, we first need to determine the area of the original triangle and then apply the scale factor.
- The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
From the information given, the original triangle has a height of 4 inches and a base of 6 inches. Therefore, the area of the original triangle is:
\[ A = \frac{1}{2} \times 6 \times 4 = \frac{1}{2} \times 24 = 12 \text{ square inches} \]
- When the triangle is dilated by a scale factor of \( k \), the area of the dilated triangle increases by a factor of \( k^2 \). In this case, the scale factor is 2.5, so we calculate \( k^2 \):
\[ k^2 = (2.5)^2 = 6.25 \]
- Now we can find the area of the dilated triangle by multiplying the original area by the factor \( k^2 \):
\[ \text{Area of dilated triangle} = \text{original area} \times k^2 = 12 \times 6.25 = 75 \text{ square inches} \]
Thus, the area of Mike’s dilated triangle is \( \boxed{75} \) square inches.