An open box is made from a square piece of cardboard 30 inches on a side by cutting identical squares and turning up the sides. A) express the volume of the box as V, as a function of the length of the side of the square cut from each corner, x. B) Find and interpert V (3), V (4), V (5), V (6), V (7). What is happening to the volume of the box as th elength of the side of the square cut from each corner increases? C) Find the domain of V

Step-by-step explanation:

A)

The length of the box is 30 - 2x inches.

The width of the box is 30 - 2x inches.

The height of the box is x inches.

So the volume is:

V = x (30 - 2x) ²

B)

V (3) = 3 (30 - 6) ² = 1728

V (4) = 4 (30 - 8) ² = 1936

V (5) = 5 (30 - 10) ² = 2000

V (6) = 6 (30 - 12) ² = 1944

V (7) = 7 (30 - 14) ² = 1792

As x increases, the volume of the box increases to a maximum and then decreases.

C)

The ends of the domain occur when V = 0.

0 = x (30 - 2x) ²

x = 0 or 15

So the domain is (0, 15).