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.
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)}.
To find the volume of the box, we need to consider the dimensions of the box after the corners are cut out and folded up.
The length and width of the base of the box will be reduced by twice the length of the cut-out squares (2x) on each side.
Therefore, the dimensions of the base of the box will be: (10 - 2x) by (6 - 2x).
The height of the box will be equal to the length of the cut-out squares, which is x.
So, the volume V of the box is given by the formula: V = length × width × height.
Substituting the values, we have:
V = (10 - 2x) × (6 - 2x) × x
Simplifying this expression will give us the volume of the box as a function of x.