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Question

An open box is to be made out of a piece of a square card board of sides 18 cm by cutting off equal squares from the corners and turning up the sides. Find the maximum volume of the box.

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Solution

Let each side of the square cut off from each corner be x cm
Then the base of the box will be of side 182x cm and the height of the box will be x cm
Then volume of box V=(182x)(182x)x
V=(182x)2x
V=4x3+324x72x2 ...(i)
Differentiating w.r t to x, we get
dVdx=12x2+324144x
dVdx=12(x212x+27) ....(ii)
For maximum volume dVdx=0
12(x212x+27)=0
x29x3x+27=0
(x9)(x3)=0
x=9,3
Again differentiating, we get
d2Vdx2=2x12 ...(iii)
At x=9,
d2Vdx2=+ve
V is minimum at x=9 at x=3
d2Vdx2=ve
V is maximum at x=3
Maximum volume V=(186)(186)×3
=12×12×3=432cm3

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