jmc

Let the height of the box be $x$. After using the Pythagorean Theorem three times, we can quickly see that the sides of the triangle are 10, $\sqrt{{\left(\frac{x}{2}\right)}^2+64}$, and $\sqrt{{\left(\frac{x}{2}\right)}^2+36}$. Since the area of the triangle is $30$, the altitude of the triangle from the base with length $10$ is $6$. Considering the two triangles created by the altitude, we use the Pythagorean theorem twice to find the lengths of the two line segments that make up the base of $10$. We find:$$10=\sqrt{\left(28+x^2/4\right)}+x/2$$ Solving for $x$ gives us $x=\frac{36}{5}$. Since this fraction is simplified:$$m+n=\boxed{41}$$