Diameter is given (log of given size), thus D is constant

$b^2 + d^2 = D^2$

$2b\dfrac{db}{dd} + 2d = 0$

$\dfrac{db}{dd} = -\dfrac{d}{b}$

Stiffness:

$k = bd^3$

$\dfrac{dk}{dd} = b(3d^2) + d^3 \dfrac{db}{dd} = 0$

$3bd^2 + d^3 \left( -\dfrac{d}{b} \right) = 0$

$3bd^2 = \dfrac{d^4}{b}$

$3b^2 = d^2$

$d = \sqrt{3} \, b$

$\text{depth } \, = \sqrt{3} \, \times \, \text{ breadth}$ *answer*