To find the area of both the original triangle and the dilated triangle, we start with the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Step 1: Calculate the area of the original triangle.
The base of the original triangle is 4 inches and the height is 6 inches.
\[ \text{Area}_{\text{original}} = \frac{1}{2} \times 4 \times 6 = \frac{1}{2} \times 24 = 12 \text{ square inches} \]
Step 2: Determine the dimensions of the dilated triangle.
Marisol dilates the triangle by a scale factor of 1.5. This means both the base and height will increase by this factor.
\[ \text{New base} = 4 \times 1.5 = 6 \text{ inches} \] \[ \text{New height} = 6 \times 1.5 = 9 \text{ inches} \]
Step 3: Calculate the area of the dilated triangle.
Now, we can use the new dimensions to find the area of the dilated triangle.
\[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times 6 \times 9 = \frac{1}{2} \times 54 = 27 \text{ square inches} \]
Step 4: Compare the areas.
We want to find out how many times larger the area of the dilated triangle is compared to the original triangle.
\[ \text{Factor of increase} = \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{27}{12} = \frac{9}{4} = 2.25 \]
Thus, the area of the dilated triangle is 2.25 times the area of the original triangle.
The final answer is:
\[ \text{The area of the dilated triangle is } 2.25 \text{ times the area of the original triangle.} \]