An open-top box is to be constructed from a 6 in by 2 in rectangular sheet of tin by cutting out squares of equal size at each corner, then folding up the resulting flaps. Let x denote the length of the side of each cut-out square. What is the volume?

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Answer 1 · 2016-06-17T10:13:10

Volume $=x^3-16x^2+12x$

Height of box: $=x$
Length of box: $=6-2x$
Width of box: $=2-2x$

Volume of box:
$x*(6-2x)(2-2x)$
$color(white)("XXX")=x*(12-16x+4x^2)$
$color(white)("XXX")=x^3-16x^2+12x$

Answer 2 · 2016-06-17T10:16:21

Volume of open-top box would be $4x^3-16x^2+12x$

As $x$ is the length of the side of each cut out square, the height of the open=top box will be $x$ .

Its length will be $(6-2x)$

and width would be $(2-2x)$

Hence volume would be

$x(6-2x)(2-2x)$

= $x(12-12x-4x+4x^2)$

= $x(12-16x+4x^2)$

= $12x-16x^2+4x^3$

or $4x^3-16x^2+12x$