A rectangular box has a base that is 4 times as long as it is wide. The sum of the height and the girth of the box is 200 feet. (a) Express the volume V of the box as a function of its width w. Determine the domain of V (w).

1. volume = 40(20 - w)

2. Domain of V (w) : (0, 20)

Step-by-step explanation:

1. Given that: l = 4w, h + G = 200 feet.

where: l is the length of the base, h is the height, G is the girth or perimeter of the rectangle, and w the width.

Girth = Perimeter = 2 (l + w)

                             = 2(4w + w)

                             = 10w

⇒ h + G = h + 10w = 200

               h = 200 - 10w ............. 1

The volume of the box can be determined by;

volume = length × width × height

            = l × w × h

            = 4w × w × (200 - 10w)

            = 40(20 - w)

The volume of the box = 40(20 - w)

2.  From the equation of the volume, 4w > 0, w > 0 and 20 - w > 0 (w> 20)

To have the box, w should not be equal to 0 or 20.

Domain of V (w) : (0, 20)