Question

If 2400 square centimeters of material is available to make a box with a square base and an open top, how do you find the largest possible volume of the box?

Answer 1

The dimension of the box I found are:
`28.3xx28.3xx14.1cm` for a volume of `11,292.5cm^3`:

Explanation:

The surface `S` of a box with open top is the sum of the surfaces of a square (base, of side `a`) and 4 rectangles (base `a` and height `h`):
`S=a^2+4a*h` (1)
while the volume `V` will be the area of the base times the height, or:
`V=a^2*h` (2)

We get `h=(S-a^2)/(4a)` from (1) put it into (2):

`V=a^2(S-a^2)/(4a)=1/(4a)(Sa^2-a^4)=S/4a-a^3/4`
maximize this volume deriving with respect to `a` and setting it equal to zero:
`(dV)/(da)=S/4-3/4a^2=0`
`3/4a^2=S/4`
`a^2=S/3`
`a=+-sqrt(2400/3)=+-28.3cm`

We use `a=+28.3cm` that in (1) gives us:
`h=(2400-28.3^2)/(4*28.3)=14.1cm`

Giving a volume of:
`V=a^2*h=11,292.5cm^3`