You are given 1200 cm^2 of cardboard to make a box with a square base and an open top. Find the largest possible volume of the box.

Answer 1

let the height be y and each side of the base be x cm
Surface area
= SA
= x^2 + 4xy = 1200
y = (1200 - x^2)/(4x)
= 300/x - (1/4)x

V= x^2 y = x^2(300/x - (1/4)x )
= 300x - (1/4)x^3
dV/dx = 300 - (3/4)x^2
= 0 for a max of V
(3/4)x^2 = 300
x^2 = 1200/3 = 400
x = √400 = 20

largest volume = 300(20) - (1/4)(8000)
= 4000 cm^3