An open box is to be made using one metre square of cardboard .find the maximum value of the box given that it has a square base

Answer 1

If x-cm cuts are made, the base is 100-2x cm, and the height is x cm.

The area is thus

(100-2x)^2 + 4x(100-2x) = 100^2
x = 2500/(100-2x) - 4(100-2x)

The volume is

v = x(100-2x)^2
= (2500/(100-2x) - 4(100-2x))(100-2x)^2

Now expand that, take dv/dx=0 and find x for maximum volume.

see the graph of v at

http://www.wolframalpha.com/input/?i=%282500%2F%28100-2x%29+-+4%28100-2x%29%29%28100-2x%29^2+for+35%3C%3Dx%3C%3D65