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

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Sum

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

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Solution

Let the side of the square cut off from the corners be x cm.

Therefore each side of the square box is (18 – 2x) cms and height is x cms.


Let V be volume of box.

∴ V = (18 – 2x)2x

∴ V = 4(9 – x2) · x

= 4(81 – 18x + x2)x

V = 4(x3 – 18x2 + 81x)

∴ `(dv)/(dx)` = 4(3x2 – 36x + 81)

= 12(x2 – 12x + 27)

= 12(x – 3)(x – 9)

Put `(dv)/(dx)` = 0,

∴ 12(x – 3)(x – 9) = 0

∴ x = 3 or x = 9

Since x = 9 is not possible,

∴ x = 3

Now `(d^2v)/(dx^2)` = 12(2x – 12)

= 24(x – 6)

\[\therefore \left. \frac{d^2V}{dx^2}\right]_{x=3}\] = 24(– 3)

= – 72 < 0

∴ V is maximum at x = 3

∴ If height of box is 3 cm, volume is maximum.

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Chapter 2.2: Applications of Derivatives - Long Answers III

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