Apiece of cardboard is 15 inches by 30 inches. a square is to be cut from each corner and the sides folded up to make an open-top box. what is the maximum possible volume of the box? round your answer to the nearest four decimal places.

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 =