A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm2) is equal to :
Explanation
Let the square of x cm
length to be cut off for the
maximum volume.
So, the length and
breadth of the box will be
(30 – 2x) and height will
be x cm.
If V be the volume of the
box, So
V = l × b × h
⇒V=x(30−2x)2
ddxd=x×2(30−2)×(−2)+(30−2)2
=(30−2)[−4x+−30−2]
∴ddxd=(30−2)(30−6)x
For maxima or minima putting dxdV=0
x=15or x=5
If x=15 thenv=o which is not possible and if x=then
dx2d2V=(30−2x)(−6)+(30−6x)(−2)
=−180+12x−60+12x=24x−240
\left. \frac{d^{2}V}{dx^{2}} \right|_{x=5} = 120 - 240 = -120 < 0
⇒V is maximum, when x is 5
⇒therefore l=b=30−2x=30−2×5=20
and h=5
⇒Surface area (without top)=2(lb+bh+hl)−lb
=2(20×20+20×5+5×20)−20×20=800 cm2
