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

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