To find the length of each side of the cubed box, we first need to determine the radius of the ball.
The formula for the surface area \( S \) of a sphere is given by:
\[ S = 4 \pi r^2 \]
We know the surface area \( S \) is 200.96 square inches. We can substitute that into the formula:
\[ 200.96 = 4 \pi r^2 \]
Next, using \( \pi \approx 3.14 \):
\[ 200.96 = 4 \times 3.14 \times r^2 \]
Calculating \( 4 \times 3.14 \):
\[ 4 \times 3.14 = 12.56 \]
So we have:
\[ 200.96 = 12.56 r^2 \]
Now, solve for \( r^2 \):
\[ r^2 = \frac{200.96}{12.56} \approx 16 \]
Taking the square root of both sides to find \( r \):
\[ r = \sqrt{16} = 4 \text{ inches} \]
The diameter \( d \) of the ball is twice the radius:
\[ d = 2r = 2 \times 4 = 8 \text{ inches} \]
Since the ball fits perfectly inside the box and touches all six sides, the side length of the cubed box is equal to the diameter of the ball.
Therefore, the length of each side of the box is:
\[ \boxed{8} \text{ inches} \]