Given that: A box open from top is made from a rectangular sheet dimension a×b by cutting squares each of side x from each of the four corners and folding up the flaps and the volume of the box is maximum.
dimensions of the box:
L=a-2x,B=b-2x&H=x
V=L×B×H
V(x)=(a−2x)(b−2x)x
V(x)=4x3−2(a+b)x2+abx
For volume to be maximum:
V′(x)=12x2−4(a+b)x+ab=0
x=244(a+b)−16a2+16b2−16ab[+cannotbetakenasx<a,b],
for maximum volume, x=6a+b−a2+b2−ab