Find the dimensions of a rectangular box without a top, of the maximum capacity with a surface area of $108 \, cm^2$.
This is my attempt at solving the problem :
If $x,y,z$ are the dimensions of the box,
Surface Area :
$$xy + 2xz + 2yz = 108\,cm^2$$
Volume :
$$V = xyz$$
Best Answer
HINT:
As $x,y,z>0$, using A.M, G.M inequality $$xy+2zx+2yz\ge 3(xy\cdot 2zx\cdot 2yz)^{\frac13}=3\cdot2^{\frac23}\cdot (xyz)^\frac23$$
Taking cube in either side, $$3^3\cdot 2^2\cdot (xyz)^2\le (xy+2zx+2yz)^3=(108)^3=2^6\cdot3^9$$
$$\implies (xyz)^2\le2^4\cdot3^6\implies xyz\le 2^2\cdot3^3=108 $$