To find the area of the original triangle, we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle with a base of 10 cm and height of 15 cm:
\[ \text{Area}_{\text{original}} = \frac{1}{2} \times 10 , \text{cm} \times 15 , \text{cm} = \frac{1}{2} \times 150 , \text{cm}^2 = 75 , \text{cm}^2 \]
Next, Kierra dilates the triangle using a scale factor of 45. The new dimensions (base and height) of the triangle after dilation will be:
\[ \text{New base} = 10 , \text{cm} \times 45 = 450 , \text{cm} \] \[ \text{New height} = 15 , \text{cm} \times 45 = 675 , \text{cm} \]
Now we can find the area of the dilated triangle:
\[ \text{Area}{\text{dilated}} = \frac{1}{2} \times \text{new base} \times \text{new height} \] \[ \text{Area}{\text{dilated}} = \frac{1}{2} \times 450 , \text{cm} \times 675 , \text{cm} \]
Calculating this area:
\[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times 303750 , \text{cm}^2 = 151875 , \text{cm}^2 \]
Now, we need to find the difference in areas between the original triangle and the dilated triangle:
\[ \text{Difference} = \text{Area}{\text{original}} - \text{Area}{\text{dilated}} \] \[ \text{Difference} = 75 , \text{cm}^2 - 151875 , \text{cm}^2 \]
As we can see, the area of the original triangle is not greater than the area of the dilated triangle; in fact, it is much smaller. Thus, the area of the original triangle is not "greater."
Instead, the correct calculation reflects that the dilated triangle is greater than the original triangle by:
\[ \text{Difference} = \text{Area}{\text{dilated}} - \text{Area}{\text{original}} = 151875 , \text{cm}^2 - 75 , \text{cm}^2 = 151800 , \text{cm}^2 \]
So, Kierra finds out that the dilated triangle is much greater in area.
Thus, the answer to the question as stated ("How much greater is the area of the original triangle than the area of the dilated triangle?") is that the original triangle is not greater; the dilated triangle is greater by 151800 cm².
This means if we were to select from the provided options, we would not find a valid answer under the assumption of the phrasing of the question.
The original question appears to misrepresent the relationship, as dilating the triangle should lead to a much larger area rather than considering it reduced in any fashion.