To find the value of \( x \), we can use the property of similar triangles, which states that corresponding sides of similar triangles are in proportion.
Let’s denote the sides of the first triangle as follows:
- Side A: 39.6 inches
- Side B: 26.4 inches
- Side C: \( x \) (the unknown side)
For the second triangle, we can assume the corresponding sides are proportional. Let's denote the corresponding side of the second triangle as \( 11.1 \) inches.
We can set up the following proportion based on the similarity of the triangles:
\[ \frac{39.6}{11.1} = \frac{26.4}{x} \]
Now, we can cross-multiply to solve for \( x \):
\[ 39.6 \cdot x = 26.4 \cdot 11.1 \]
Calculating the right side:
\[ 26.4 \cdot 11.1 = 293.04 \]
So, we have:
\[ 39.6 \cdot x = 293.04 \]
Now, divide both sides by \( 39.6 \):
\[ x = \frac{293.04}{39.6} \]
Calculating the value of \( x \):
\[ x \approx 7.4 \]
Thus, the value of \( x \) rounded to the nearest tenth is \( \boxed{7.4} \) inches.