Answer
$\left(\sqrt 3-\frac{1}{2}\right)$ m
Work Step by Step
The distance from corner to corner of the box, D, can be found with pythagorean theorem;
$\sqrt{1^2 + 1^2 + 1^2} = \sqrt3$ meters
Now we find the distance from corner to corner in terms of the radius of the spheres, r. First, finding the distance from a corner to the center of the nearby sphere (Lc), which can again be found using pythagorean theorem;
$L_{c} = \sqrt{r^2 + r^2 + r^2} = \sqrt{3r^2} = r\sqrt3$
Now imagine a line going from the center of a corner sphere to the center of the opposing corner sphere's center. This line would pass through the center of the center sphere, and travel the radii of both corner spheres. This gives us a length:
$L_{l} = 2r + r + r = 4r$
We can use these to give us the total length, from one corner to the oppposing one, in terms of r:
$L = 4r + r\sqrt3 + r\sqrt3 = r(4+ 2\sqrt3)$
This length is equal to the number we found initially, and setting them equal to each other will let us solve for r.
$\sqrt3 = r(4 + 2\sqrt3)$
$r=\frac{\sqrt 3}{4+2\sqrt 3}=\frac{\sqrt 3(4-2\sqrt 3)}{(4+2\sqrt 3)(4+2\sqrt 3)}=\frac{4\sqrt 3-6}{4}=\sqrt 3-\frac{1}{2}$
