If 1800 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Question

If 1800 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

 

Volume=___?

Answer 1

An open top for the box gives all 1800 square centimeters of material

to the four "walls" and the "floor" of the box. 

Build the Equation For Surface Area as: 

Surface Area Equals [Length Times Length] (since the base is a square)

Plus [4 Times Length Times Height].

Rewrite Surface Area as A = L² + 4LH.

Build the Equation For Volume as:

Volume = Height Times [Length Times Length].

Rewrite Volume as HL².

Now equate 1800 to L² + 4LH and write {1800 − L²}/4L = H.

Rewrite H as 450/L − L/4.

This renders Volume equal to:

V = L²[450/L − L/4] or 450L − L³/4.

Take dV/dL as 450 − (3/4)L².  

Equate 450 − (3/4)L² to 0 which forces L²

to 600 and L to √600.

Then take the Maximum Volume sought as 

450√600 − (√600)³/4 which goes to

7348.469228 cubic centimeters. 

Taking d²V/dL² or (-6/4)L and calculating -6/4 × √600 gives -1.5√600, 

which is less than 0 and indicates a maximum volume at L = √600

as opposed to a minimum.