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
and how would you graph this

Answer 1

I think I understand your question. Do you mean that the flat piece of material is 4 X 6, and you are cutting squares out of the corners to fold it into a box? If so, then

V = l X w X h

l = 6 - 2x
w = 4 - 2x
h = x

so: V = 4x^3 -20x^2 + 24x

Answer 2

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

Graphing is probably easiest with a graphing calculator, but if you had to graph by hand, find the zeros (y = 0 when x = 0, 2, or 3) and figure out the direction of the curves between these points and their relative maximum and minimum values - this will involve taking a derivative of this and doing some analysis, which is a little time consuming but still entirely possible.

Answer 3

Volume = (4-2X) * (6-2X) * x
Volume = (4(x^3)) - (20(X^2)) + 24X
Cubic Function when graphed.

If you are in Calculus there is a way to find out what the maximum volume would be by Optimization, edit your question if you are.