Mensuration CMI 2014

Let $x,y,z$ be the side lengths of this rectangular box. The total side length computes to $4(x+y+z) = 60 \Rightarrow x+y+z = 15$ with the maximum volume occurring for the cube $x = y = z = 5$ . Hence, our required box volume range is $\boxed{(0,125]}.$

Given the total surface area is $2(xy + xz + yz) = 56$ (i) and that $x+y+z = 15$ (ii), the largest diagonal can be found from squaring (ii) to produce:

$(x+y+z)^2 = 15^2;$

or $(x+y)^2 + 2(x+y)z + z^2 = 225$ ;

or $x^2 + y^2 + z^2 + 2(xy + xz + yz) = 225;$

or $x^2 + y^2 + z^2 = 225 - 56$ ;

or $\sqrt{x^2 + y^2 + z^2} = \sqrt{169};$

or $\sqrt{x^2 + y^2 + z^2} = \boxed{13}.$

For the box volume, $V = xyz$ , it's not possible to determine since we were originally given two equations, (i) and (ii), in three unknowns. Altogther, Choice A is correct.