Question 932918
The length (in cm) is {{{sqrt(4^2+6^2+12^2)+sqrt(16
36+144)=sqrt(196)=highlight(14)}}} .
 
THE EXPLANATION:
Let's say the box is sitting on one of the rectangular sides that measure 6 cm by 12 cm.
We call that the base of the box.
The diagonal splits that base into 2 right triangles with legs measuring 6 cm by 12 cm.
The length of that diagonal, {{{green(d)}}} cm, according to the Pythagorean theorem, is such that
{{{green(d)^2=6^2+12^2}}} .
{{{drawing(300,150,-2,13,-1,6.5,
rectangle(0,0,12,6),rectangle(0,0,0.5,0.5),
green(line(0,6,12,0)),
locate(0.05,3.2,6cm),locate(5.8,0.7,12cm),
locate(5.8,3,green(d))
)}}}
That diagonal is perpendicular to one of the vertical 4-cm long edges of the box.
The longest possible stick, of length {{{red(s)}}} cm, stick will form a right triangle with the diagonal of the base and a vertical edge.
This is what the (see through) box looks like:
{{{drawing(360,180,-2,16,-1,8,
rectangle(0,3,12,7),triangle(14,0,0,3,12,3),
line(2,0,14,0),line(2,4,14,4),line(0,7,12,7),
line(2,0,2,4),line(14,0,14,4),line(0,3,0,7),
line(0,3,2,0),line(12,7,14,4),line(0,7,2,4),
green(line(0,3,14,0)),red(line(0,3,14,4)),
locate(-0.3,1.8,6cm),locate(7.8,0,12cm),
locate(7,1.5,green(d)),locate(8.5,3.8,red(s)),
locate(14.05,2.5,4cm)
)}}}
According to the Pythagorean theorem,
{{{red(s)^2=4^2+green(d)^2}}}
{{{red(s)^2=4^2+6^2+12^2}}}
{{{red(s)=sqrt(4^2+6^2+12^2)}}}