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Suppose you want to make an open box out of a piece of card board by cutting small squares at the four corners and folding up the sides. If the piece of card board is a square whose sides are 1 m. long, how big a square should you cut from the corners to maximize the volume of the box?

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Answer:

The box of length
x=(1)/(6) must be cut from each corner to maximize the volume.

Explanation:

It is given that the side of a square card board is 1 m.

Let small squares of length x are cut from the four corners. The dimensions of the box are

Length = (1-2x) m

Width = (1-2x) m

Height = x m

The volume of a cuboid is


V=l* b* h

where, l is length and b is breadth and h is height.

The volume of a rectangular box.


V=(1-2x)(1-2x)x


V=(1-2x)(x-2x^2)


V=x-2x^2-2x(x-2x^2)


V=x-2x^2-2x^2+4x^3


V=4x^3-4x^2+x

We need to maximize the volume. Differential above equation with respect to x.


V'=12x^2-8x+1 .... (1)

Equate V'=0 to find the critical points.


12x^2-8x+1=0


12x^2-6x-2x+1=0


6x(2x-1)-1(2x-1)=0


(6x-1)(2x-1)=0


x=(1)/(6),(1)/(2)

Differential equation (1) with respect to x.


V''=24x-8

At
x=(1)/(6)


V''=24((1)/(6))-8=-4<0

Volume is maximum at
x=(1)/(6).

At
x=(1)/(2)


V''=24((1)/(2))-8=4>0

Volume is minimum at
x=(1)/(6).

Therefore the box of length
x=(1)/(6) must be cut from each corner to maximize the volume.

User Danial Kosarifa
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