FInding a function that represents the volume of a box?

Question

example
the box is 4 x 6 foot
you cut out equal squares on each corner
let x denote thelength of each side of the square to be cut out

Answer 1

If the box is 4 by 6, and you cut corners so that you can fold the sides up and end up with an open-top box, then you would have the following:

height of the box = x
length of the box = 6 - 2x (one x for each side you cut)
width of the box = 4 - 2x (one x for each side you cut)

The volume then is the product of the three:

V = x(6 - 2x)(4 - 2x)
V = x(24 - 20x + 4x^2)
V = 24x - 20x^2 + 4x^3

Answer 2

Volume = V
width = 4 - 2X
length = 6 - 2x
hight = x

V = L.W.H
V = (-2x + 6)(-2x + 4)(x)
V = (4x^2 -20x + 24)(x)
V = 4x^3 -20x^2 +24x

If you are trying to find out the biggest volume the box can be, graph the function f(x) = 4x^3 -20x^2 +24x and find its maximum point. Then simply find the x coordinate which is the value for x.

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Answer 3

The corner cuts must be square, say x by x. The finished length and width will be 2x short. The volume is

(L - 2x)(W - 2x)