See attachment

The volume of the box will be:2. Equation for the quantity being maximized.

V=a (16-2a) (30-2a)

This will be a solution of:3. Dimensions of the square resulting in the maximum volume.

dV/da=0.

dV/da=d/da[4a^3 - 92a^2 + 480a]

.........=12 a^2 - 184 a + 480

So we want the roots of:

12 a^2 - 184 a + 480=0,

which are a=10/3, and a=12.

The second of these roots is clearly non-physical, which leaves the first

To check if this is a maximum we need to check that d^2V/da^2 is negative.

d^2V/da^2 = 24 a - 184

This is <0 when a=10/3.

so we conclude that a=10/3 gives a maximum volume for the box.

RonL