Answer:
Maximum volume = 649.519 cubic inches
Step-by-step explanation:
A rectangular piece of cardboard of side 15 inches by 30 inches is cut in such that a square is cut from each corner. Let x be the side of this square cut. When it was folded to make the box the height of box becomes x, length becomes (30-2x) and the width becomes (15-2x).
Volume is given by
V = 
First, we differentiate V(x) with respect to x, to get,

Equating the first derivative to zero, we get,

Solving, with the help of quadratic formula, we get,
,
Again differentiation V(x), with respect to x, we get,

At x =
,

Thus, by double derivative test, the maxima occurs at
x =
for V(x).
Thus, largest volume the box can have occurs when
.
Maximum volume =