The areas of the three adjacent faces meeting at one vertex of a cuboid are $4000 \, cm^2$,$2000 \, cm^2$,and $3200 \, cm^2$ respectively. Find the dimensions and the volume of the cuboid.

(A) Let the dimensions of the cuboid be $l, b,$ and $h$. The areas of the three adjacent faces are given by $lb = 4000$,$bh = 2000$,and $hl = 3200$.
Multiplying these three equations: $(lb)(bh)(hl) = 4000 \times 2000 \times 3200$.
$(lbh)^2 = 25,600,000,000$.
$lbh = \sqrt{25,600,000,000} = 160,000 \, cm^3$.
Now,to find the dimensions:
$l = (lbh) / (bh) = 160,000 / 2000 = 80 \, cm$.
$b = (lbh) / (hl) = 160,000 / 3200 = 50 \, cm$.
$h = (lbh) / (lb) = 160,000 / 4000 = 40 \, cm$.
Thus,the dimensions are $80 \, cm, 50 \, cm, 40 \, cm$ and the volume is $160,000 \, cm^3$.