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Question

A box, constructed from a rectangular metal sheet, is 21 cm by 16 cm by cutting equal squares of side x cm from the corners of the sheet and then turning up the projected portions. The value of x (in cm) so that volume of the box is maximum is

A
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B
2
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C
3
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D
4
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Solution

The correct option is C 3
The dimensions of the box after cutting equal squares of side x on the corner will be 212x,162x, and height x.
V=x(212x)(162x)
=x(33674x+4x2)=4x3+336x74x2

dVdx=12x2+336148x
dVdx=012x2+336148x=03x237x+84=0(x3)(3x28)=0x=3,283
For x=283162x<0
Which is not possible,
x=3
d2Vdx2=6x37d2Vdx2x=3=<0
So for x=3 will give maximum volume.

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