When a pentagon (or any polygon) is dilated by a scale factor \( k \), the lengths of its sides change by that same scale factor \( k \). If \( k < 1 \), then the lengths of the sides are reduced.
The relationship between the area of a polygon and the scale factor is that the area changes by the square of the scale factor. Specifically, the area \( A' \) of the dilated polygon is given by:
\[ A' = k^2 \cdot A \]
Where \( A \) is the original area of the polygon. Since \( k < 1 \), \( k^2 < 1 \) as well, meaning that the area \( A' \) will decrease.
Now, considering how the area decreases in relation to the decrease in the lengths of the sides, since the lengths decrease by a factor of \( k \), the comparison between the decrease in area and the decrease in side lengths can be summarized as follows:
- Area decreases by a factor of \( k^2 \).
- Lengths decrease by a factor of \( k \).
Since \( k^2 < k \) (because \( k < 1 \)), the area decreases more rapidly than the lengths of the sides decrease.
Thus, the correct answer is:
D. The area decreases more rapidly than the lengths of the sides decrease.