Volume Function

Question

An open box is to be made from a flat piece of material 10 inches long and 5 inches wide by cutting equal squares of length x from the corners and folding up the sides.
Write the volume Vof the box as a function of x. Leave it as a product of factors, do not multiply out the factors.

V =

If we write the domain of the box as an open interval in the form (a,b), then what is a =?

a =

and what is b =?

b =

Answer 1

Let c = 10 and d = 5.  We cut squares of length x from the corners, then

 

the length of the box is l = c - 2x,

 

the width of the box is w = d - 2x, and

 

the height of the box is h = x.

V = lwh = (c - 2x)(d - 2x)(x) = (10 - 2x)(5 - 2x)x

The volume of the box is positive and it is not possible to cut squares of lengths that are more than d/2 or 5/2.  So the domain for the volume or the possible values of x is (0,5/2).  a = 0 and b = 5/2.