from a square piece of board of area 144 m² . an open box of square base and 2 metres in height is made. find the volume of the box and area of its wall
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Asked on October 15, 2019 by
Kabita Mathew
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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ANSWER
Let each side of the square cut off from each corner be x cm
Then the base of the box will be of side 18−2x cm and the height of the box will be x cm
Then volume of box V=(18−2x)(18−2x)x
V=(18−2x)
2
x
V=4x
3
+324x−72x
2
...(i)
Differentiating w.r t to x, we get
dx
dV
=12x
2
+324−144x
dx
dV
=12(x
2
−12x+27) ....(ii)
For maximum volume
dx
dV
=0
⇒12(x
2
−12x+27)=0
⇒ x
2
−9x−3x+27=0
⇒ (x−9)(x−3)=0
⇒ x=9,3
Again differentiating, we get
dx
2
d
2
V
=2x−12 ...(iii)
At x=9,
dx
2
d
2
V
=+ve
∴ V is minimum at x=9 at x=3
dx
2
d
2
V
=−ve
∴ V is maximum at x=3
∴ Maximum volume V=(18−6)(18−6)×3
=12×12×3=432cm
3
Answer:
Volume: 128 m2
Area of its walls: 64 m2