Using the least amount of material(calculus related)?

Question

If 1200 cm^2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Answer 1

Side = X
Height = h

Surface area = X^2 + 4 xh = 1200
h = (1200 - x^2)/4x

Volume = x^2*h
V = x^2 *(1200 - x^2)/4x
V = X(1200 - X^2)/4
V = 300X - X^3/4
dV/dX = 300 - 3 X^2/4
x= 20
Substitue -> h = 10

V = 400*10 = 4000 cm^3

Answer 2

let the side of the base be x and the volume be V and the height=h
area of the box=x^2+4hx=1200 4hx=1200-x^2 and h=1200-x^2/4x
Volume
hx^2
V=hx^2=[(1200-x^2)/4x]*x^2=> x(1200-x^2)/4
dv/dx=1/4[-2x+1200-x^2]=0 for max
-x^2-2x+1200=0 => x^2+2x-1200=0
solve for 'x' and substitute to get the maximum volume