Answer:
(a) See attached
(b) 3 m by 4 m by 2 m, has least surface area
Step-by-step explanation:
You want the width and area of a 2 m high box that has a volume of 24 m³, given its length is a whole number of meters, and you want dimensions that minimize the cost of the box.
Width
The volume is given by ...
V = LWH
We are given the volume and the height, and asked to find the width for various lengths.
24 = LW·2
W = 12/L . . . . . . . solve for W
The values of width for the various lengths are shown in the attachment.
Area
The surface area of the box is given by ...
A = 2(LW +H(L +W))
We know that LW = 12, so this becomes ...
A = 2(12 +2(L +W)) = 24 +4(L +W)
The values of L+W are 13, 8, 7, and the corresponding areas are 76, 56, 52 square meters. These are shown in the table in the attachment.
Cost
We presume the main driver of the cost of the box is its surface area, so the cost will be minimized when the area is minimized. This will be the case for a box with dimensions that are either of ...
3 m long by 4 m wide by 2 m high
4 m long by 3 m wide by 2 m high
