To find the value of \( x \) that maintains the same proportion between the large rectangle and the small rectangle, we can set up a proportion using the given dimensions.
The large rectangle dimensions are:
- Length = 16.8 feet
- Width = 2.3 feet
The smaller rectangle dimensions are:
- Length = 4.5 feet
- Width = \( x \) feet
We will use the ratio of the length and width of both rectangles to find \( x \).
Set up the ratio for the rectangles:
\[ \frac{\text{Length of large rectangle}}{\text{Width of large rectangle}} = \frac{\text{Length of small rectangle}}{\text{Width of small rectangle}} \]
Inserting the values gives us:
\[ \frac{16.8}{2.3} = \frac{4.5}{x} \]
Now, we can cross-multiply to solve for \( x \):
\[ 16.8 \cdot x = 2.3 \cdot 4.5 \]
Calculate \( 2.3 \cdot 4.5 \):
\[ 2.3 \cdot 4.5 = 10.35 \]
So, the equation becomes:
\[ 16.8x = 10.35 \]
Now, divide both sides by 16.8 to isolate \( x \):
\[ x = \frac{10.35}{16.8} \]
Calculating this gives:
\[ x \approx 0.61607 \]
Rounding this to the nearest tenth results in:
\[ x \approx 0.6 \]
Thus, the value of \( x \) rounded to the nearest tenth is:
\[ \boxed{0.6 \text{ feet}} \]