Question 825568
The volume of the box, V = l*w*h
Since the height, h is fixed at 10 cm, the problem involves varying the length and width to maximize the area.
Since the perimeter = 2(l + w) = 42 -> w = 21 - l
Now we have an expression for A in terms of the single variable, l:
A = l(21 - l) = 21l - l^2
The area will be maximized where dA/dl = 0
dA/dl = 21 - 2l = 0 -> l = 21/2 = 10.5
Therefore w = 21 - 10.5 = 10.5, i.e., the volume is maximized when the bottom is a square.  The total volume, V = 10.5^2*10 = 1102.5 cm^3