Let the side of the square to be cut off be xcm. Then, the length and breadth of the box will be (30−2x)cm each and the height of the box is xcm therefore,
The volume V(x) of the box is given by V(x)=x(30−2x)2
dv
dx
=(30−2x)2+2x×(30−2x)(−2) 0=(30−2x)2−4x(30−2x) 0=(30−2x)[(30−2x)−4x] 0=(30−2x)(30−6x) x=15,5 x≠15 (Not possible) {∴V=0} (Not possible) Surface area without top of the box =ℓb+2(bh+hℓ) =(30−2x)(30−2x)+2[(30−2x)x+(30−2x)x] =[(30−2x)2+4{(30−2x)x}. =[(30−10)2+4(5)(30−10)] =400+400 =800cm2