An open-top box is to be made out of a piece of cardboard measuring 6 {m} \times 6 {m} by cutting off equal squares from the corners and turning up the sides. The height of the box for maximum volume is

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NIMCET 2009 Question Paper with solutions

Question : 38 of 120

Marks: +1, -0

An open-top box is to be made out of a piece of cardboard measuring

6 {m} × 6 {m}

by cutting off equal squares from the corners and turning up the sides. The height of the box for maximum volume is

Solution:

Let

x

be the length of side of square cut off from the cardboard.
So, volume,

V=x[6-2 x]^{2}

....(i)
For maximum volume,

\;{d V}/{d x}=(6-2 x)^{2}+24-2 x(6-2 x)

=(6-2 x)(6-6 x)=0

⇒ \;\; x=3,1

\;{d^{2} V}/{d x^{2}}=(6-2 x)(-6)+(6-6 x)(-2)

=-36+12 x-12+12 x

=24 x-48

\;\; x=1, \;{d^{2} V}/{d x^{2}}<0

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