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

Surface area of box=1200 cm²

Volume of box=s²h

s = side of square base

h = height of box

S. A. = s² + 4sh

S. A. = surface area or 1200 cm², s²

= the square base, and 4sh = the four 'walls' of the box.

1200 = s² + 4sh

1200 - s² = 4sh

(1200 - s²) / (4s) = h

v (s) = s² ((1200 - s²) / (4s))

v (s) = s (1200 - s²) / 4.

v (s) = 300s - (1/4) s^3

by derivating

v' (s) = 300 - (3/4) s²

0 = 300 - (3/4) s²

-300 = (-3/4) s²

400 = s²

s = - 20 and 20.

again derivating

v" (s) = - (3/2) s

v" (-20) = - (3/2) (-20)

v" (-20) = 30

v" (20) = - (3/2) (20)

v" (20) = - 30

v (s) = 300s - (1/4) s^3

v (s) = 300 (20) - (1/4) (20) ^3

v (s) = 6000 - (1/4) (8000)

v = 6000 - 2000

v=4000