Question

A manufacturer wants to make a box with a volume of 24 cubic feet.

Answer 1

First: You should know the formulas for volume and surface area of a rectangular prism.
$V=whl$
$SA=2wl+2wh+2hl$

Now, if our volume is 24 ft³, then the 3 dimensions of the box must multiply to make 24. We want to make two designs, so let's think of some possible dimensions.
Let's say...2 × 2 × 6 and 2 × 3 × 4.
You should be able to sketch boxes and label the sides. Just draw two rectangles and draw lines between all of the points. (see attached for reference)

Now all we have to do is find the surface area. Plug in our width, length, and height and solve.
First box
SA = 2wl + 2wh + 2hl
SA = 2×2×2 + 2×2×6 + 2×2×6
SA = 8 + 24 + 24
SA = 56
Second box
SA = 2wl + 2wh + 2hl
SA = 2×2×3 + 2×2×4 + 2×3×4
SA = 12 + 16 + 24
SA = 52

At $4 / ft²...
56 ft² = $224
52 ft² = $208

The second box is less expensive to produce because it has less surface area and therefore less wood needed to build the sides of the box.