Gby cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box can be made. if the cardboard is 17 in. long and 10 in. wide and the square cutaways have dimensions of x in. by x in., find a function giving the volume v(x) of the resulting box.

We find the function by starting from the formula of the volume of a rectangular box.

Volume of a rectangular box = length x width x height

From the problem statement,
length = 17 - 2x
width = 10 - 2x
height = x

where x is the height of the box or the side of the equal squares from each corner and turning up the sides

V = (17-2x) (10-2x) (x)
V = (17 - 2x) (10x - 2x^2)
V = 170x - 34x^2 - 20x^2 + 4x^3

V = 4x^3 - 54x^2 + 170x <THE FUNCTION V(x)