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=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.