An open-top box is to be constructed from a 6 foot by 8 foot rectangular cardboard by cutting out equal squares at each corner and the folding up the flaps

please see the attached file.

1) An open-top box is to be constructed from a 6 foot by 8 foot rectangular cardboard by cutting out equal squares at each corner and the folding up the flaps. Let x denote the length of each side of the square to be cut out.

a) Find the function V that represents the volume of the box in terms of x.

Volume is equal to length x width x height. The height will be x. One side will be equal to 6 - 2x, and the other will be equal to 8 - 2x. Therefore, the volume can be calculated as:

V = (x)(6 - 2x)(8 - 2x)

This can be simplified as:

V = 4x(3 - x)(4 - x)

Notice that x has to be greater than 0 (you can't have a negative height, and if x = 0, you just have a flat piece of cardboard, not a box), and x has to be less than 3 (2x can't be larger than one of the sides).

b) Graph this function and show the graph over the valid range of the variable x.

Remember that x has to be between 0 and 3 (0 < x < 3). Over that range of x, the graph looks like this:

c) Using the graph, what is the value of x that will produce the maximum volume?

There is a maximum at x = 1.131. The maximum volume is V = 24.268.