A box open from top is made from a rectangular sheet of dimension a× b by cutting squares each of side x from each of the four corners and folding up…

Given that: A box open from top is made from a rectangular sheet dimension $a\times b$a×b by cutting squares each of side $x$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

$$V=L\times B\times H$$V=L×B×H

$$V(x)=(a-2x)(b-2x)x$$V(x)=(a−2x)(b−2x)x

$$V(x)=4{x}^{3}-2(a+b){x}^{2}+abx$$V(x)=4x3−2(a+b)x2+abx

For volume to be maximum:

$${V}^{'}(x)=12{x}^{2}-4(a+b)x+ab=0$$V′(x)=12x2−4(a+b)x+ab=0

$x=\frac{4(a+b)-\sqrt{16{a}^{2}+16{b}^{2}-16ab}}{24}$x=244(a+b)−16a2+16b2−16ab​​$[+\mathrm{cannot}\mathrm{be}\mathrm{taken}\mathrm{as}x<a,b]$[+cannotbetakenasx<a,b],

for maximum volume, $x=\frac{a+b-\sqrt{{a}^{2}+{b}^{2}-ab}}{6}$x=6a+b−a2+b2−ab​​