See diagram below:

Let S be the surface area

Let V be the volume of the box

We have S = 1200

=> 1200 = x^2 + 4xy .........see diagram

=> y = 300/x - x/4

Now V = x^2 * y

=> V = x^2 (300/x - x/4)

=> V = 300x - (x^3)/4

=> V' = 300 -(3/4)x^2

For max V, set V' = 0

=> 300 -(3/4)x^2 = 0

=> (3/4)x^2 = 300

=> x^2 = 400

=> x = +/- 20

=> x = 20, since x cannot be negative

but y = 300/x - x/4

=> y = 300/20 - 20/4

=> y = 10

so for max volume, the dimensions are x = 20, y = 10

since V = x^2 * y

max V = (20)^2 * 10

=> maxV = 4000 cm^3