Question

Mai wants to make an open-top box by cutting out corners of a square piece of cardboard and folding up

the sides. The cardboard is 10 centimeters by 10 centimeters. The volume in cubic centimeters of the open-
top box is a function of the side length in centimeters of the square cutouts.
Which of the following represents the volume of the box V(x).

Answer 1

The volume V(x) of Mai's open-top box, created by cutting out corners with side length x from a 10 cm by 10 cm square piece of cardboard and folding up the sides, is V(x) = 4
`x^3 - 20x^2` + 100x cubic centimeters.

Step-by-step explanation:

To determine the volume of Mai's open-top box, we start with a 10 cm by 10 cm square piece of cardboard. When a square of side length x is cut from each corner, and the sides are folded up, the new dimensions of the box will be:

Length: 10 - 2x cm

Width: 10 - 2x cm

Height: x cm

The volume V(x) of an open-top box can be calculated by multiplying these three dimensions, that is:

V(x) = (10 - 2x)(10 - 2x)x

Or more simply:

V(x) = x(100 - 20x + 4
`x^2`)

V(x) = 4
`x^3 - 20x^2` + 100x

So, the function representing the volume of the box V(x) as a function of the side length of the square cutouts x is:

V(x) =
`4x^3 - 20x^2` + 100x