I see this problem in 2 steps.

Firstly a 2D problem looking down onto the top of the box. From here you a maximising a rectangle inside a semi circle.

You need to find the area of this rectangle as a function and solve it when its derivative is equal to zero.

If the rectangle in question has dimensions y and x then \(\displaystyle A(x,y) = 2xy\) and \(\displaystyle r^2 = x^2+y^2\) then \(\displaystyle A(x) = 2x\times \sqrt{14.5^2-x^2}\) now find \(\displaystyle x\) where \(\displaystyle A'(x)=0\)

After you have this we can find the third deminsion.