A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 6 in. by 10 in. by cutting out equal squares of side x at each corner and then folding up the sides as shown in the figure. Express the volume V of the box as a function of x.

Answer 1

We start by cutting out squares of side x at each corner of the rectangular piece of cardboard. This will leave us with a rectangular base of length 10-2x and width 6-2x.

Next, we fold up the sides to create the open-top box. The height of the box will be equal to the side of the squares we cut out, which is x.

Therefore, the volume V of the box can be expressed as the product of the length, width, and height:
V = (10-2x)(6-2x)(x). Answer: \boxed{(60x - 32x^2 + 4x^3)}.